Projectile Motion

Projectile model and trajectories

Projectile motion treats a launched body as a particle subject only to uniform gravity. With horizontal $x$ and upward $y$ axes,

a_{x} = 0, a_{y} = - g .

Launch speed $v_{0}$ at angle $θ$ has initial components

v_{0 x} = v_{0} cos θ, v_{0 y} = v_{0} sin θ .

Integrating each independent component gives

x = x_{0} + v_{0} cos θ t,

y = y_{0} + v_{0} sin θ t - \frac{1}{2} g t^{2} .

Time is the shared parameter. Horizontal motion runs at constant velocity, vertical motion at constant acceleration. Eliminating time between the two equations gives the parabolic path.

The horizontal component $v_{0} cos θ$ stays fixed for the whole ideal flight; the vertical component $v_{0} sin θ$ decreases linearly under gravity.

The launch arrow is decomposed at the release point. Horizontal and vertical components set the initial data for independent kinematic equations; their vector sum is the specified launch velocity.

Inverse design for nonlevel targets.

A forward calculation starts from launch speed and angle and returns a landing point. Inverse design starts from a target coordinate and solves for launch conditions that reach it. A target coordinate alone does not fix a unique trajectory: height, range, obstacle clearance, flight time, and available speed each add a constraint.

For a fixed target $(R, H)$ and chosen launch angle $θ$ , the trajectory equation solves for the required launch speed:

v_{0}^{2} = \frac{g R ^{2}}{2 cos ^{2} θ ( R tan θ - H )} .

The denominator must be positive: the initial straight-line rise $R tan θ$ must exceed the target height $H$ , so the launch direction points above the target before gravity bends the path down to it. A direction pointing below the target has no real solution in the ideal model.

Fixed-time design.

A specified arrival time $T$ determines the initial components directly:

v_{0 x} = \frac{R}{T}, v_{0 y} = \frac{H + \frac{1}{2} g T ^{2}}{T} .

The required speed and angle follow from

v_{0} = v_{0 x}^{2} + v_{0 y}^{2}, θ = atan2 (v_{0 y}, v_{0 x}) .

Each positive $T$ gives one ideal-model solution. A short delivery time needs a large horizontal component; a long one needs a larger upward component to stay aloft.

Landing-surface constraints.

A real landing surface is rarely an infinite plane. A platform can occupy a finite horizontal interval, cap the impact speed, or demand a specific impact angle. Apply these conditions after finding candidate intersections. For a platform from $x_{1}$ to $x_{2}$ , a root is admissible only when $x_{1} \leq x (t_{f}) \leq x_{2}$ . An impact-speed limit uses

v_{f}^{2} = v_{0}^{2} - 2 g (y_{f} - y_{0})

in the ideal model. An impact-angle requirement uses

tan ϕ = \frac{v _{y} ( t _{f} )}{v _{x} ( t _{f} )} .

A geometrically reachable target can still fail: the projectile may arrive too fast, at too shallow an angle, or outside the finite landing region.

On a sloped surface, compare this velocity angle with the local surface angle, not with horizontal alone, to get the incidence angle.

The launch velocity is resolved into horizontal and vertical components. Gravity changes only the vertical component, so equal-time trajectory points have equal horizontal spacing while their vertical separation changes with time.

Trajectory equation.

The horizontal equation gives $t = (x - x_{0}) / (v_{0} cos θ)$ when the horizontal component is nonzero. Substitution in the vertical equation yields

y - y_{0} = (x - x_{0}) tan θ - \frac{g ( x - x _{0} ) ^{2}}{2 v _{0}^{2} cos ^{2} θ} .

The negative quadratic coefficient curves the trajectory downward. Use this form when the question specifies a horizontal location, such as a barrier height or a target position. It fails once drag, lift, or a large variation of gravity enters, because the component accelerations are then no longer constant.

Equal-time samples expose the two component motions. Horizontal increments stay equal because $a_{x} = 0$ . Vertical increments shrink on ascent, reach zero velocity (not zero acceleration) at the top, and grow on descent.

At equal launch and landing heights, horizontal velocity is unchanged and vertical velocity reverses sign. Thus

v_{x, f} = v_{0} cos θ, v_{y, f} = - v_{0} sin θ .

Impact speed then equals launch speed in the ideal model, but the velocities differ: horizontal components agree while vertical components point in opposite directions. Equal speed is not equal velocity.

Range and launch geometry

Vertical velocity is $v_{y} = v_{0} sin θ - g t$ . The highest point has $v_{y} = 0$ , which gives

t_{H} = \frac{v _{0} sin θ}{g}, H - y_{0} = \frac{v _{0}^{2} sin ^{2} θ}{2 g} .

At the peak, vertical velocity is zero but vertical acceleration remains $- g$ . Equal launch and landing heights give the flight time $T = 2 v_{0} sin θ / g$ , so the range is

R = v_{0} cos θ T = \frac{v _{0}^{2} sin ( 2 θ )}{g} .

Complementary launch angles have the same ideal range because they give the same value of $sin (2 θ)$ . They do not have the same flight time or maximum height.

The vertical-velocity line crosses zero at peak time. Equal endpoint heights produce equal positive and negative velocity magnitudes, while the fixed negative slope equals the constant gravitational acceleration throughout flight.

The velocity-time graph fixes the peak time but hides the spatial trade-off between horizontal and vertical components. Complementary trajectories share a range while differing in height and flight duration.

Complementary launch angles yield the same ideal range on level ground. The steep trajectory has a greater height and longer flight, while the shallow trajectory has a larger horizontal component; their common endpoint gives equal range.

Launch and landing at unequal heights.

The equal-height range relation does not apply when the landing surface lies above or below the launch point. The vertical equation remains a quadratic in time,

\frac{1}{2} g t^{2} - v_{0} sin θ t + (y_{f} - y_{0}) = 0.

Its roots are

t = \frac{v _{0} sin θ \pm v _{0}^{2} sin ^{2} θ - 2 g ( y _{f} - y _{0} )}{g} .

The stated landing event selects the physical root. A target above launch height is crossed twice, ascending and descending. A lower target usually has one positive future root and one belonging to the trajectory extrapolated before launch. Only after choosing the time does horizontal displacement follow: $x_{f} - x_{0} = v_{0} cos θ t_{f}$ .

A launch from an elevated platform reaches a lower landing surface at the positive root of the vertical quadratic. The dashed level marks release height; the vertical drop increases flight time and range relative to level ground.

Design constraints and launch platforms

On level ground with fixed launch speed and no drag,

R (θ) = \frac{v _{0}^{2}}{g} sin (2 θ) .

Differentiation gives

\frac{d R}{d θ} = \frac{2 v _{0}^{2}}{g} cos (2 θ) = 0,

so $θ = 4 5^{\circ}$ maximizes range in the physical interval.¹ This holds only for equal launch and landing heights; a target, barrier, time, or peak-height constraint shifts the optimum. Complementary angles give equal range on level ground but differ in flight time and peak height.

Launch from an elevated platform.

A launch point above the landing plane changes the flight-time equation. With horizontal distance $x$ and vertical coordinate measured from the landing plane,

x = v_{0} cos θ t, 0 = h + v_{0} sin θ t - \frac{1}{2} g t^{2} .

The physically relevant root is

t_{f} = \frac{v _{0} sin θ + v _{0}^{2} sin ^{2} θ + 2 g h}{g} .

The negative root places the projectile at the landing height before launch and does not represent the forward flight. The range is

R (θ) = \frac{v _{0} cos θ}{g} (v_{0} sin θ + v_{0}^{2} sin^{2} θ + 2 g h) .

The expression reduces to the level-ground formula when $h = 0$ . For $h > 0$ , the range-maximizing angle is less than $4 5^{\circ}$ : the added fall time allows a larger horizontal launch component.

A $20.0 m$ platform and $v_{0} = 12.0 m s^{- 1}$ horizontal launch give

t_{f} = \frac{2 h}{g} = 2.02 s, R = v_{0} t_{f} = 24.2 m .

The impact velocity components are $v_{x} = 12.0 m s^{- 1}$ and $v_{y} = - g t_{f} = - 19.8 m s^{- 1}$ . Thus

v_{f} = v_{x}^{2} + v_{y}^{2} = 23.2 m s^{- 1}, θ_{f} = atan2 (∣ v_{y} ∣, v_{x}) = 58. 8^{\circ} .

The impact angle is steeper than the launch angle because gravity has changed only the vertical component.

Range optimization as a constrained calculation.

For equal launch and landing heights, differentiating

R (θ) = \frac{v _{0}^{2}}{g} sin (2 θ)

gives

\frac{d R}{d θ} = \frac{2 v _{0}^{2}}{g} cos (2 θ) .

The stationary condition $cos (2 θ) = 0$ gives $θ = 4 5^{\circ}$ in the interval from $0$ to $9 0^{\circ}$ . The second derivative is negative there:

\frac{d ^{2} R}{d θ ^{2}} = - \frac{4 v _{0}^{2}}{g} sin (2 θ) < 0,

so the stationary point is a maximum. This result holds only while launch speed is fixed and both heights match. A target at a specified horizontal range has two possible launch angles when the range is less than the maximum. The lower-angle trajectory has shorter flight time and a smaller peak; the higher-angle trajectory has larger vertical speed and may clear an obstacle.

A minimum-speed problem reverses the constraint. For a target at $(R, H)$ , set the trajectory equation equal to $H$ and solve for $u = tan θ$ :

\frac{g R ^{2}}{2 v _{0}^{2}} u^{2} - R u + (H + \frac{g R ^{2}}{2 v _{0}^{2}}) = 0.

The discriminant determines whether a launch angle exists. Its zero value marks the smallest speed that reaches the target. This quadratic structure is more reliable than memorizing a nominal $4 5^{\circ}$ rule.

Air resistance as a model change.

Drag breaks the constant-horizontal-velocity equation. A common high-speed approximation is

F_{D} = - \frac{1}{2} C_{D} ρ A v v .

Acceleration now depends on both speed and direction. The path stops being a parabola, complementary angles lose their equal ranges, and equal-height launch and impact speeds differ. Numerical integration replaces the closed-form equations, with the no-drag solution as a baseline for sizing the drag effect.

Launches from moving platforms.

The initial ground-frame velocity is a Galilean vector sum. A platform with velocity $V_{P / G}$ and launch velocity $v_{B / P}$ measured on the platform gives

v_{B / G} (0) = v_{B / P} + V_{P / G} .

After release, gravity acts exactly as for a stationary launcher. The platform velocity shifts the initial ground-frame components, not the acceleration. For a platform moving horizontally at $V$ and a forward launch of speed $u$ at angle $α$ relative to the platform,

v_{0 x} = V + u cos α, v_{0 y} = u sin α .

The ground-frame trajectory is therefore

y - y_{0} = \frac{u sin α}{V + u cos α} (x - x_{0}) - \frac{g ( x - x _{0} ) ^{2}}{2 ( V + u cos α ) ^{2}} .

The platform frame assigns different initial components to the same motion. A frame change is consistent only when target coordinates and velocities transform together with the projectile velocity.

Interception equates the projectile and target positions. A target with constant ground velocity $v_{T}$ satisfies

r_{B} (0) + v_{B / G} (0) t - \frac{1}{2} g t^{2} \overset{}{^} = r_{T} (0) + v_{T} t .

Both components must hold at one positive time. A fixed launch speed then admits zero, one, or two launch directions, depending on target position and motion.

Target geometry, feasibility, and checks

With horizontal displacement $R$ and height $H$ relative to launch, the target equation can be written in terms of $u = tan θ$ :

H = R u - \frac{g R ^{2}}{2 v _{0}^{2}} (1 + u^{2}) .

Rearrangement gives a quadratic equation for the launch-slope variable,

\frac{g R ^{2}}{2 v _{0}^{2}} u^{2} - R u + (H + \frac{g R ^{2}}{2 v _{0}^{2}}) = 0.

Its discriminant controls reachability:

Δ = R^{2} - \frac{2 g R ^{2} H}{v _{0}^{2}} - \frac{g ^{2} R ^{4}}{v _{0}^{4}} .

If $Δ > 0$ , two launch angles strike the target. If $Δ = 0$ , one limiting angle reaches it. If $Δ < 0$ , the specified speed is insufficient. The two-angle case generalizes complementary angles on level ground: one path is lower and faster, the other higher and slower. The discriminant is the direct mathematical reachability test.

For fixed speed, solving the target equation for $H$ as a function of $R$ identifies every point reachable at a selected launch angle. The upper boundary of the union of those trajectories is the range envelope. Maximizing the height at each $R$ gives

H_{env} = \frac{v _{0}^{2}}{2 g} - \frac{g R ^{2}}{2 v _{0}^{2}} .

Targets above this downward-opening parabola cannot be reached at the stated speed in the ideal model. Targets below it have one or two possible launch angles, except where a ground or platform constraint removes one path.

The range envelope bounds all drag-free trajectories launched with one fixed speed from the origin. A target below the envelope admits two launch angles; a target on it has one limiting tangent trajectory, and a target above it is unreachable without increasing speed or changing the launch point.

Energy cross-check of speed.

Gravity is conservative in the no-drag model. Between launch and any point of height $y$ , mechanical energy gives

\frac{1}{2} m v^{2} + m g y = \frac{1}{2} m v_{0}^{2} + m g y_{0} .

Hence

v^{2} = v_{0}^{2} - 2 g (y - y_{0}) .

Use this relation to check the speed magnitude. Component equations determine the velocity angle and horizontal location at a stated time. At equal launch and landing heights, energy gives equal launch and impact speeds, matching the component result. At a lower landing height, energy predicts greater impact speed; the vertical kinematic equation determines the impact direction.

Numerical model with drag.

Quadratic drag changes the acceleration to

a = - g \overset{}{^} - \frac{C _{D} ρ A}{2 m} v v .

The drag vector opposes the instantaneous velocity. Its horizontal component is nonzero whenever the horizontal speed is nonzero, so the constant- $v_{x}$ equation is lost. An Euler or midpoint update advances the state over short time steps:

v_{n + 1} = v_{n} + a_{n} Δ t, r_{n + 1} = r_{n} + v_{n} Δ t .

Step-size convergence provides an error estimate. Repeat the calculation with $Δ t$ and $Δ t /2$ ; the difference in range, peak height, or intercept time estimates the numerical sensitivity. This does not measure model error from an unknown drag coefficient, wind variation, or spin-induced lift. Those uncertainties must be varied separately in the input model.

Trajectory reconstruction and data fitting

The projectile equations describe the center of mass after release. A real launcher adds pre-release geometry (nozzle height, rail direction, platform motion, the first unobstructed point), and a real landing surface adds its own (a plane, a finite deck, a curved surface, a target of nonzero size). The kinematic solution is complete only after the intersection with that geometry is tested.

A landing plane through point $(x_{p}, y_{p})$ with slope $q$ obeys

y - y_{p} = q (x - x_{p}) .

Substitution of the projectile trajectory gives a quadratic in $x$ . Its roots are the potential intersections of the ideal path and the plane. A physical landing requires a future time, a point within the finite material extent of the surface, and a velocity directed into rather than away from the contact surface. These three tests separate mathematical curve intersections from actual collision events.

The local normal to a plane of slope $q$ is proportional to $(- q, 1)$ . At an intersection, the incoming velocity component normal to the surface is

v_{n} = \frac{- q v _{x} + v _{y}}{1 + q ^{2}} .

The sign identifies whether the projectile approaches the selected surface face. The tangential component determines sliding or impact-angle conditions in a later contact model; it is not determined by projectile kinematics alone.

Fitting a trajectory to data.

Video or motion-tracking data can test the projectile model without directly measuring launch angle. With a calibrated image plane, record pairs $(t_{i}, x_{i}, y_{i})$ . The ideal model predicts

x_{i} = x_{0} + v_{0 x} t_{i},

y_{i} = y_{0} + v_{0 y} t_{i} - \frac{1}{2} g t_{i}^{2} .

A linear fit of $x$ against $t$ estimates $v_{0 x}$ . A quadratic fit of $y$ against $t$ estimates $y_{0}$ , $v_{0 y}$ , and the coefficient of $t^{2}$ . The expected quadratic coefficient is $- g /2$ . A statistically significant deviation can indicate timing error, camera perspective distortion, air resistance, or an incorrect spatial calibration.

The fit residuals have diagnostic value. Random residuals of comparable size can be consistent with measurement noise. Residuals that become increasingly negative late in the flight indicate that the observed path falls below the ideal parabola, a common drag signature. Residuals with a left-right trend can indicate a wind or camera-axis error.

Air-resistance scale analysis.

Quadratic drag has magnitude $D = \frac{1}{2} C_{D} ρ A v^{2}$ . A dimensionless estimate of its importance over a ballistic arc compares drag with weight,

Π_{D} = \frac{C _{D} ρ A v _{0}^{2}}{2 m g} .

When $Π_{D} ≪ 1$ , drag is initially small compared with weight, though it can still accumulate over a long flight. When $Π_{D}$ is comparable with one, the ideal parabolic prediction lacks quantitative reliability. Numerical integration is then required because drag direction changes with velocity. Use the ratio to select the model before calculation.

Interpreting fitted launch data.

A fit constrains physical parameters only once its outputs are tied back to the launcher. The intercepts $x_{0}$ and $y_{0}$ describe the chosen coordinate origin, not the physical edge of the launcher. The fitted components $v_{0 x}$ and $v_{0 y}$ describe the center of mass at the instant after contact is lost. When the first camera frame falls after release, extrapolate back to $t = 0$ : the first visible data point is not automatically the initial condition.

Fit parameters should be reported with the measurement role that produced them. This keeps a camera coordinate, a launch state, and a model residual from being collapsed into one apparent trajectory result.

Fitted or measured item	Representation	Diagnostic
Image position	calibrated $(x_{i}, y_{i}, t_{i})$	scale, perspective, and timestamp checks
Horizontal motion	$x = x_{0} + v_{0 x} t$	residual trend versus time
Vertical motion	$y = y_{0} + v_{0 y} t - \frac{1}{2} g t^{2}$ in the ideal model	fitted quadratic coefficient
Departure from ideal flight	residual vector	drag, wind, spin, or camera-model test

Camera-based fitting requires a spatial calibration. A known ruler, target spacing, or calibrated reference plane converts image pixels to metres. A camera tilted relative to the motion plane introduces perspective distortion: equal physical horizontal distances may not occupy equal image distances. A two-dimensional fit is therefore valid only after lens distortion and projective geometry are negligible or corrected. High frame rate improves timing resolution, but it does not repair an incorrect spatial calibration.

Measurement and model inference use separate stages:

Establish a coordinate origin and a metric scale in the motion plane.
Record a time for each measured center-of-mass position.
Fit horizontal position to a line and vertical position to a quadratic.
Inspect residuals before interpreting fitted launch components.
Compare the fitted vertical quadratic coefficient with $- g /2$ .
State whether deviations are consistent with uncertainty, drag, wind, spin, or camera geometry.

Residuals should be plotted against time rather than reported only as one aggregate error number. A constant offset can arise from an origin choice and does not by itself invalidate the model. A curvature mismatch indicates a physical or timing model error. A sequence of residuals that alternates sign can indicate frame-by- frame tracking noise. The pattern matters because different errors require different corrections.

Reporting a projectile result.

A technical projectile result states the coordinate convention, launch point, initial velocity (components or speed and angle), gravitational value, target or landing geometry, and assumptions about air forces and frame motion. Omitting any of these leaves the result ambiguous: a range without launch and landing heights, an angle without its reference direction, or a speed without its reference frame.

The following compact report form contains the minimum information required for reproduction:

r_{0} = (x_{0}, y_{0}), v_{0} = (v_{0 x}, v_{0 y}), a = (0, - g),

followed by the event condition, such as $y = 0$ , $x = R$ , or intersection with a specified surface. If uncertainty is material, report it with the corresponding quantity and method: measurement standard deviation, calibration bound, or model parameter range. This notation distinguishes observed inputs from derived outputs.

Assumption hierarchy and communication limits.

Present the ideal model as an approximation hierarchy. A report should state that uniform gravity was used, air resistance was neglected, the launch platform was stationary in the ground frame, the trajectory remained in a vertical plane, and the target coordinate was measured relative to the launch point. Each condition identifies a possible source of departure between calculation and experiment.

When results are extrapolated beyond measured data, the extrapolation should be named. A fit over the first half of a flight does not validate the landing point if drag grows with speed or if a crosswind changes later. A camera fit in one plane does not establish the absence of out-of-plane motion. Good reporting separates what was measured, what was inferred from a model, and what was predicted outside the measurement interval.

Initial-state measurements determine the component equations; the acceleration model determines the trajectory family; and event geometry selects a physical solution from that family. Omitting any link can change the stated range or impact condition without an obviously implausible curve, so projectile calculations must document assumptions alongside numerical outputs.

A projectile report is a short chain from measurement to prediction, one model quantity per stage:

Stage	Mathematical object	Required check
Launch state	$(x_{0}, y_{0}, v_{0 x}, v_{0 y})$	reference frame, time origin, and spatial calibration
Trajectory model	constant $g$ or a stated drag law	residual pattern against measured positions
Event condition	target point, line, or region	selected root and its physical time interval
Reported outcome	range, impact speed, or angle	model domain and uncertainty source

Reproducibility checklist.

Before comparing two projectile calculations, align their definitions. The same launch can be reported with horizontal range, along-slope range, map displacement, or distance travelled; these are different quantities. The same target can be specified by a fixed point, a moving point, a line, or a finite region. The same velocity can be measured relative to ground, launcher, air, or water. Each choice changes the equations used to define the event, even when the drawn trajectory appears similar.

A reproducible solution records the event time selected from any quadratic roots, the sign convention for vertical coordinates, and the unit system used in numerical substitution. It also distinguishes a feasibility conclusion from a launch-angle solution: a nonnegative discriminant establishes that a solution exists, while the roots determine the actual candidate directions. These reporting details prevent coordinate and reference-frame errors from being hidden by a plausible final range.

Inclined and fixed-speed trajectories

An inclined landing surface changes the intersection condition. Place the launch point at the origin and let the plane rise at angle $β$ above horizontal. Its equation is

y = x tan β .

The projectile trajectory for launch angle $θ$ is

y = x tan θ - \frac{g x ^{2}}{2 v _{0}^{2} cos ^{2} θ} .

Equating the two curves gives the nonzero horizontal intersection coordinate,

x = \frac{2 v _{0}^{2} cos ^{2} θ}{g} (tan θ - tan β) .

The expression is meaningful only when the projectile initially rises above the plane, requiring $θ > β$ for an upward-sloping plane. The distance measured along the plane is $s = x / cos β$ , so

s = \frac{2 v _{0}^{2} cos ^{2} θ}{g cos β} (tan θ - tan β) .

An equivalent trigonometric form is

s = \frac{2 v _{0}^{2} cos θ sin ( θ - β )}{g cos ^{2} β} .

Unlike the level-ground range, this result depends on the plane angle explicitly. The landing point is lower or higher than launch according to the sign of the plane slope, and the distance convention must be stated: horizontal range $x$ and slope distance $s$ are different observables.

Optimization on a slope.

Maximizing $s$ requires differentiation of the inclined-plane expression rather than setting $θ = 4 5^{\circ}$ . The condition can be written in angle form as

\frac{d}{d θ} [cos θ sin (θ - β)] = 0.

Expansion or trigonometric identities yield

tan (2 θ - β) = cot β,

which gives the physical maximum

θ_{max} = 4 5^{\circ} + \frac{β}{2} .

An upward-sloping plane has an optimum launch angle steeper than $4 5^{\circ}$ . A downward slope has a shallower optimum. The result concerns distance measured along the plane and assumes fixed launch speed, no drag, and a plane passing through the launch point.

Envelope of fixed-speed trajectories.

Fix the launch point and speed $v_{0}$ , but permit the launch angle to vary. Every angle produces one parabola, and the family has an upper boundary called the trajectory envelope. It identifies the highest point reachable at each horizontal coordinate and therefore gives a direct obstacle-clearance criterion.

Start with the trajectory equation and write $u = tan θ$ . Since $sec^{2} θ = 1 + u^{2}$ ,

y = xu - \frac{g x ^{2}}{2 v _{0}^{2}} (1 + u^{2}) .

At fixed horizontal coordinate $x$ , maximize the right side with respect to the launch-slope variable. Differentiation gives

\frac{\partial y}{\partial u} = x - \frac{g x ^{2}}{v _{0}^{2}} u = 0,

so the angle of the trajectory tangent to the envelope satisfies

u = tan θ = \frac{v _{0}^{2}}{g x} .

Substitution back into the family of trajectories yields

y_{env} = \frac{v _{0}^{2}}{2 g} - \frac{g x ^{2}}{2 v _{0}^{2}} .

The envelope is itself a downward-opening parabola. Its intercept with the vertical axis is $v_{0}^{2} / (2 g)$ , the maximum vertical height attainable by a straight-up launch. Its intercept with the launch-height line occurs at $x = v_{0}^{2} / g$ , which is half the maximum level-ground range. This does not mean that the envelope is one physical trajectory: each point on it is tangent to a different member of the fixed-speed trajectory family.

The heavy curve is the envelope of every drag-free trajectory launched from one point at fixed speed. A barrier below the envelope can be cleared by at least one angle; a barrier on it has one tangent limiting launch; a barrier above it cannot be reached without increasing speed or changing launch conditions.

Clearance of an obstacle of height $H$ at horizontal location $R$ requires

H \leq \frac{v _{0}^{2}}{2 g} - \frac{g R ^{2}}{2 v _{0}^{2}} .

The inequality is equivalent to the nonnegative-discriminant condition in the target-angle quadratic: the obstacle must lie on or below the envelope. The equality case is sensitive to small changes in speed, launch position, drag, and obstacle height. Physical design therefore requires an additional safety margin.

Solution methods and feasibility domains.

Identify the unknown before selecting equations. Four common problem forms cover most questions:

Forward trajectory: launch conditions are known; calculate position or velocity at a stated time.
Landing event: a surface or target defines the final position; solve the vertical equation for physically admissible time, then use horizontal motion.
Inverse design: target coordinates and an angle, speed, or time constraint are known; solve for the remaining launch quantity.
Feasibility test: speed and target geometry are fixed; use the discriminant or trajectory envelope before attempting a numerical angle.

The equations should be written in a fixed order. First establish the coordinate origin and signs. Then resolve initial velocity. Next write one equation per component with the same time variable. Finally apply the event condition. Eliminating time too early often hides which root represents the intended event; retaining time until the event is identified makes the physical branch visible.

A projectile meeting a vertical wall at $x = R$ has the single horizontal-motion time

t_{R} = \frac{R}{v _{0} cos θ} .

Substitution in vertical motion gives its wall height. For a horizontal platform at height $H$ , vertical motion gives one or two candidate times. Each time is then tested in the horizontal interval occupied by the platform. A wall and a platform therefore impose different mathematical constraints even when they pass through the same point.

Measurement and uncertainty in projectile tests.

Experimental projectile data are sensitive to launch-angle and speed uncertainty. For level-ground range,

R = \frac{v _{0}^{2} sin ( 2 θ )}{g},

small independent uncertainties give the first-order estimate

\frac{δ R}{R} \approx 2 \frac{δ v _{0}}{v _{0}} + 2 cot (2 θ) δ θ .

Angle uncertainty is measured in radians in this differential expression. Near $4 5^{\circ}$ , $cot (2 θ)$ approaches zero, so range is locally insensitive to small angle errors; at shallow or steep angles, the same angular error produces a larger fractional range change. Speed uncertainty always enters with a factor of two because range scales with speed squared.

Feasibility and limiting cases.

Limiting cases test the domain of a result. As $g$ tends to zero, the trajectory equation approaches a straight line. As $v_{0}$ tends to zero, nonzero-range targets become unreachable. At $θ = 9 0^{\circ}$ , the horizontal range equation is undefined because horizontal velocity is zero, but vertical motion remains well defined. At $θ = 0$ , the maximum-height expression gives zero rise above launch point. These limits test the physical domain of compact formulas and prevent division-by-zero expressions from being used as general laws.

Assumptions and validation

Four assumptions underlie the standard equations, each removing a term from the acceleration model:

Particle body: no rotation, no spin lift, no deformation.
Uniform gravity: constant magnitude and vertical direction, giving $a_{y} = - g$ .
Inertial ground frame over the flight interval.
No air forces: drag and lift neglected, giving $a_{x} = 0$ .

The trajectory is a parabola only while all four hold together.

The model is often accurate for short terrestrial trajectories of compact objects at modest speed. It becomes inaccurate when flight time is long, projectile speed is large, altitude change is comparable with Earth radius, or aerodynamic forces are comparable with weight. A calculation should state the model before reporting a range or an impact speed. A numerical answer without its acceleration model is not fully specified.

Timing constraints and event selection.

The vertical equation can generate more than one mathematical time. A target at a height between launch level and peak height may be crossed on ascent and again on descent. The physical event determines which root is used. A negative time belongs to an extrapolated trajectory before the stated launch and must be rejected for a forward-time prediction.

At a target $(R, H)$ , horizontal motion gives $t = R / (v_{0} cos θ)$ when the horizontal component is positive. Substitution in vertical motion produces the target equation. This approach is preferable when target coordinates are known, because it makes the single shared time explicit. Solving horizontal and vertical motion with different independently chosen times describes no physical intersection.

A horizontal target line below the peak intersects the same trajectory twice. The earlier point is an ascending crossing and the later point is a descending crossing; both share the same target height but have different times and vertical-velocity signs.

Horizontal and vertical initial conditions.

A horizontal launch has the special condition $v_{0 y} = 0$ . A vertical launch has the special condition $v_{0 x} = 0$ , for which the horizontal coordinate is constant and the trajectory degenerates from a parabola to a vertical line. General-angle launch contains both cases as limiting values.

θ = 0 ⟹ v_{0 y} = 0, θ = 9 0^{\circ} ⟹ v_{0 x} = 0.

The trajectory equation divides by $cos θ$ , so it cannot be used directly for a purely vertical launch. The original component equations remain valid. This is a mathematical-domain check: a compact formula can have a restricted range even when the underlying physical model remains well defined.

Error checks for numerical solutions.

A solution should satisfy independent checks. Units in the trajectory equation must reduce to length. At $t = 0$ , the position and velocity must reproduce their initial values. Differentiating the position equations must recover the velocity components, and differentiating again must give $(0, - g)$ . At equal heights, the computed speed should agree with launch speed under the no-drag model. A solution for a higher target requires a discriminant that is nonnegative. These checks catch sign errors before a numerical angle or range is interpreted.

In a numerical drag calculation, the ideal solution gives a limiting benchmark: as the drag coefficient approaches zero, the numerical range and trajectory should approach the analytic parabolic result. Halving the time step should change reported range by less than the desired numerical tolerance. Agreement between two step sizes does not validate the drag coefficient, wind model, or aerodynamic area; it only tests discretization error for the selected model.

The separation between the two curves measures the effect of changing the acceleration model. It grows during flight because drag continuously removes horizontal and vertical speed. Range, peak height, impact speed, and flight time must all be recomputed after drag enters; no single reduction factor adjusts them together.

Consistency across solution forms.

Three ideal-model calculations describe the same event through different variables:

Vertical kinematics: time at a specified height.
Trajectory equation: height at a specified horizontal coordinate.
Energy: speed magnitude at a specified height.

Agreement among them is a strong check. Disagreement usually signals a sign-convention mismatch, a wrong root, or a level-ground formula applied to a nonlevel event.

For example, at equal launch and landing heights, component kinematics gives $v_{y, f} = - v_{y, 0}$ and $v_{x, f} = v_{x, 0}$ . Squaring and adding yields $v_{f}^{2} = v_{0}^{2}$ . Mechanical energy gives the same result immediately because $Δ y = 0$ . Neither argument implies equal velocity vectors: the vertical component reverses direction. This distinction is essential when an impact angle or a surface-normal speed is required.

An inverse calculation can be checked in the opposite direction. After computing a required launch speed or angle, substitute it into the forward component equations and evaluate the specified target event. The target coordinates, time, and any stated impact constraint must all be recovered within the precision of the input measurements. This forward substitution is preferable to trusting an algebraic rearrangement alone, especially near a feasibility boundary where roundoff and parameter uncertainty can change the number of real trajectories.

Tipler and Mosca, Physics for Scientists and Engineers, 6th ed., §3-2. ↩