Projectile Motion Calculator

About Projectile Motion

This calculator models ideal projectile motion: a point mass launched with speed v₀ at angle θ from height h₀, under constant gravitational acceleration g and no air resistance. The horizontal and vertical motions are independent: horizontal speed stays constant while vertical speed changes linearly with time due to gravity.

• Components: v_x=v₀cosθ, v_y=v₀sinθ.
• Time to apex: t_apex=v_y/g.
• Max height: h_max=h₀+v_y²/(2g).
• Flight time: solve h₀ + v_yt - (g/2)t² = 0 ⇒ t_flight = (v_y + √(v_y² + 2gh₀))/g.
• Range: R=v_x·t_flight.

Real trajectories in air are shorter and lower due to drag; this ideal model matches most classroom exercises.

Projectile Motion Explained (Formulas, Assumptions & Examples)

Projectile motion describes the path of an object launched with an initial speed and angle under a constant downward acceleration due to gravity. In the idealized classroom model, we ignore air resistance, treat the projectile as a point mass, and assume a flat landing surface at the chosen reference level. Under these assumptions, the horizontal and vertical motions are independent: horizontal velocity stays constant while vertical velocity changes linearly with time.

Core quantities

• Inputs: launch speed v₀, angle θ (degrees), initial height h₀, gravity g.
• Components: v_x = v₀cosθ, v_y = v₀sinθ.
• Time to apex: t_apex = v_y / g (if v_y > 0).
• Max height: h_max = h₀ + v_y² / (2g).
• Flight time (from height h₀): solve h₀ + v_y t − (g/2)t² = 0 ⇒ t_flight = ( v_y + √( v_y² + 2 g h₀ ) ) / g.
• Range: R = v_x · t_flight.
• Impact speed: |v| = √( v_x² + (v_y − g t_flight)² ).

Assumptions & common mistakes

• No air drag: real trajectories are shorter and lower; this model targets standard coursework.
• Angle in degrees: convert to radians for manual calculations (the tool handles this for you).
• Gravity sign: use positive g as a magnitude; downward direction is handled in the equations.
• Initial height: when h₀>0, the range and flight time increase compared to level ground.
• Symmetry: with h₀=0 and no air, angles θ and 90°−θ yield the same range if speed is the same (classic 30°/60° result).

Quick formula list (level ground, h₀=0)

• t_flight = 2 v₀ sinθ / g
• R = (v₀² sin(2θ)) / g (maximum at θ = 45°)
• h_max = v₀² sin²θ / (2g)

Worked example

Suppose a ball is launched at v₀ = 20 m/s, θ = 45°, from ground level on Earth (g = 9.80665 m/s²). Components: v_x=14.142 m/s, v_y=14.142 m/s. Time to apex: t_apex ≈ 1.44 s. Maximum height: h_max ≈ 10.2 m. Time of flight: t_flight ≈ 2.88 s. Range: R ≈ 40.8 m. Your tool reproduces these numbers instantly and also plots the trajectory for intuition.

Tips for using the calculator

• Use the gravity preset (Earth, Moon, Mars) to explore how g changes the arc and range.
• Try a non-zero initial height to simulate launches from a platform or cliff.
• Switch speed units (m/s, km/h, mph) for quick real-world comparisons.
• Copy result rows and share deep links for assignments or lab reports—everything runs locally for privacy.