Projectile Motion Calculator

Related Calculators

Projectile motion: 2D motion where the only acceleration is gravity (no air resistance), so sideways motion and up-down motion can be handled separately.

Initial speed (m/s): The total launch speed right at the start (before splitting into sideways and vertical parts).

Launch angle: The direction you launch, measured from the horizontal. Positive angles point upward; negative angles point downward.

Gravity (m/s^2): The downward acceleration used in the model. Enter it as a positive number (Earth standard gravity is 9.80665 m/s^2) [1].

Initial horizontal speed (v_x0): The sideways part of the launch speed. With no air resistance, this stays constant.

Initial vertical speed (v_y0): The up-down part of the launch speed. It decreases by g each second.

Total time of flight (T): How long until the projectile reaches the landing level (y = 0) after launch, if it ever does.

Horizontal range (R): How far sideways it travels by the time it reaches the landing level (y = 0).

Maximum height (y_max) above landing level: The highest y-value reached, measured relative to the landing level y = 0.

Impact direction angle (degrees): The direction of the velocity right at landing, measured above the horizontal. It is usually negative because the projectile is moving downward at impact.

Target horizontal distance (m): The sideways distance you want the projectile to land from the start; used to solve for launch angle solution(s) [1].

Mistake: Entering an angle in radians while the calculator is set to degrees (for example 0.78 instead of 45).
Fix: Either switch Advanced Angle unit to Radians, or keep Degrees and type the angle in degrees.

Mistake: Typing gravity as a negative number because gravity points downward.
Fix: Enter Gravity (m/s^2) as a positive magnitude (like 9.80665). The equations already treat gravity as downward.

Mistake: Getting Total time of flight (T) as N/A and assuming the calculator is broken.
Fix: Your setup never reaches the landing level y = 0 (for example, you start below it with too little upward speed). Increase Initial speed (m/s) , increase the upward angle, or change Initial height (m) .

Mistake: In target-angle mode, using 0 or a negative Target horizontal distance (m) .
Fix: Use a positive distance. If you need a backward shot, that is a different sign convention and may not be supported by your class problem.

Mistake: Entering a snapshot time after the projectile has already landed and expecting a real height and speed.
Fix: Keep Time for snapshot (s) between 0 and Total time of flight (T) . For landing values, read the impact outputs instead.

Mistake: Mixing up what height is measured from.
Fix: This tool reports Maximum height (y_max) above landing level (relative to y = 0). If you want max height above the launch point, compute y_max minus Initial height (m) .

No air resistance: The calculator assumes the projectile does not slow down in the air. Real objects often have shorter range and a different path, especially when they are fast, light, or not very streamlined.

Constant gravity: Gravity is treated as a constant value everywhere along the path. This is a good approximation for typical classroom distances, but not for very long-range motion.

Landing level definition: Results are based on reaching the landing level y = 0. The Initial height (m) input is measured above that level, so choose your reference carefully.

2D only: Motion sideways-left/right out of the page (3D), wind, spin, and lift are not included.

Angle units and sign convention: The math depends on whether the angle is in degrees or radians (Advanced Angle unit) and on the convention that up is positive. A wrong unit setting can make every result wrong.

Feasibility limits: In target-angle mode, some distances are impossible for the chosen Initial speed (m/s), Initial height (m), and Gravity (m/s^2), so there may be 0, 1, or 2 valid angles [1].

Snapshot time boundary: Snapshot outputs are only meaningful for times between 0 and Total time of flight (T); times after landing are reported as N/A instead of continuing the equations past impact.

We use the standard no-air-resistance projectile model: horizontal motion has constant speed, and vertical motion has constant downward acceleration g [3].

Angle conversion (if using degrees): θ (radians) = θ (degrees) * π/180

v_x0 = v0 * cos(θ)

v_y0 = v0 * sin(θ)

Vertical position (up is positive, and g is entered as a positive number):

y(t) = y0 + v_y0*t - (g*t^2)/2

Time of flight T to reach the landing level y = 0 comes from solving y(T) = 0. This leads to a discriminant check:

D = v_y0^2 + 2*g*y0

If D < 0: no real T (the projectile never reaches y = 0)

If D >= 0: T = (v_y0 + sqrt(D)) / g

Once T is known:

Range: R = v_x0 * T

Time to peak: t_peak = max(0, v_y0/g)

Maximum height: y_max = y(t_peak)

Velocity at time t (used for both snapshots and impact at t = T):

v_x(t) = v_x0

v_y(t) = v_y0 - g*t

speed(t) = sqrt(v_x(t)^2 + v_y(t)^2)

direction(t) = atan2(v_y(t), v_x(t))

Target-angle mode solves for launch angle(s) that land at x = target_range and y = 0. Using u = tan(θ), the equation becomes a quadratic that can give 0, 1, or 2 real solutions [1].

A = g*x^2/(2*v0^2)

Solve for u: A*u^2 - x*u + (A - y0) = 0, where x = target_range

Then θ = atan(u) (convert to degrees for display)

Mini-example (trajectory mode): let v0 = 50 m/s, launch angle = 45 degrees, y0 = 0 m, and g = 9.80665 m/s^2 (Earth standard gravity) [1]. Then v_x0 = 35.36 m/s, v_y0 = 35.36 m/s, T = 7.208 s, y_max = 63.71 m, and R = 254.84 m (rounded).