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Projectile Motion Calculator

Enter launch speed, angle and height to get time of flight, maximum height and range with every kinematics equation shown. Free physics tool for AP Physics 1, GCSE, IB and JEE.

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Projectile results

ux = 14.142 m/s
uy = 14.142 m/s
Time of flight T = 2.886 s
Time to max height = 1.443 s
Max height H = 10.204 m
Range R = 40.816 m
Impact speed = 20 m/s
Impact angle = 45° below horizontal

Trajectory (axes not to equal scale)

H = 10.204 mR = 40.816 m
  1. 1

    Resolve the launch velocity into components

    Horizontal velocity stays constant (no air resistance); vertical velocity changes at rate −g.

    ux=ucos⁡θ=20cos⁡45∘=14.142 m/s,uy=usin⁡θ=20sin⁡45∘=14.142 m/su_x = u\cos\theta = 20\cos 45^\circ = 14.142\ \text{m/s},\qquad u_y = u\sin\theta = 20\sin 45^\circ = 14.142\ \text{m/s}
  2. 2

    Time to reach maximum height

    At the top the vertical velocity is zero: 0 = u_y − g t.

    tup=uyg=14.1429.8=1.443 st_{\text{up}} = \frac{u_y}{g} = \frac{14.142}{9.8} = 1.443\ \text{s}
  3. 3

    Maximum height

    Use v_y² = u_y² − 2gΔy with v_y = 0, then add the launch height.

    H=h+uy22g=0+(14.142)22(9.8)=10.204 mH = h + \frac{u_y^2}{2g} = 0 + \frac{(14.142)^2}{2(9.8)} = 10.204\ \text{m}
  4. 4

    Time of flight

    Launched from ground level, the flight is symmetric: y = u_y t − ½ g t² = 0 gives t = 0 or t = 2u_y / g.

    T=2uyg=2(14.142)9.8=2.886 sT = \frac{2u_y}{g} = \frac{2(14.142)}{9.8} = 2.886\ \text{s}
  5. 5

    Horizontal range

    Horizontal motion has constant velocity, so distance = u_x × time of flight.

    R=uxT=(14.142)(2.886)=40.816 mR = u_x T = (14.142)(2.886) = 40.816\ \text{m}
  6. 6

    Impact velocity

    Vertical velocity on landing is v_y = u_y − gT; combine with u_x using Pythagoras.

    vy=14.142−(9.8)(2.886)=−14.142 m/s,v=ux2+vy2=20 m/s,α=tan⁡−1 ⁣∣vy∣ux=45∘v_y = 14.142 - (9.8)(2.886) = -14.142\ \text{m/s},\quad v = \sqrt{u_x^2 + v_y^2} = 20\ \text{m/s},\quad \alpha = \tan^{-1}\!\frac{|v_y|}{u_x} = 45^\circ

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FAQ

Which angle gives maximum range?

From ground level and without air resistance, 45° gives the maximum range R = u²/g.

Is air resistance included?

No — the calculator uses ideal projectile motion with constant g.

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