Science · High school

Projectile motion

Type your own speed, angle, height and gravity: work out the range, the highest point and the time of flight, and watch the path.

In the learning section below the figure, predict, try, complete the tasks and check yourself with short questions.

Fig. 1Projectile motion

Change the speed, angle, height or gravity; press “Launch” to watch the throw

R = 35.3 m · H = 5.1 m

Time t2.04 s

Launch speed v₀20 m/s

Angle θ30°

Launch height h₀0 m

Gravity g: 9.81 m/s²

Your own values

v₀

h₀

Speed up to 300 m/s, height up to 1,000 m, g up to 300 m/s². Angle between 0° and 90°. A value applies when you press Enter or leave the box.

Results of the throw

Range R: 35.3 m
Highest point H: 5.1 m
Time of flight T: 2.04 s
Time to the top: 1.02 s
Horizontal speed vₓ (constant): 17.32 m/s
Initial vertical speed: 10 m/s
Impact speed: 20 m/s

At t = 2.04 s

Horizontal position x: 35.3 m
Height y: 0 m
Vertical speed v_y: -10 m/s
Speed: 20 m/s

Formulas used (no air resistance)

vₓ = v₀ cos θ, v_y = v₀ sin θ − g t
x = vₓ t, y = h₀ + v₀ sin θ · t − ½ g t²
T = (v₀ sin θ + √(v₀² sin² θ + 2 g h₀)) ÷ g, R = vₓ T
H = h₀ + (v₀ sin θ)² ÷ (2g); with h₀ = 0, R = v₀² sin 2θ ÷ g

Learn

Progress is kept on this page only; no points are given.

1Predict
2Try
3Explain

A ball is launched from the ground at the same speed, once at 30° and once at 60°. Which one lands farther away?

Without air resistance the horizontal and vertical motions are independent: constant speed sideways, constant acceleration up and down. Together they trace a parabola. For a real ball air resistance shortens the range and pulls the best angle below 45°; this model leaves it out. Planetary g values from NASA’s planetary fact sheets, rounded. Galileo, Two New Sciences (1638), fourth day.

Projectile motion

Results of the throw

At t = 2.04 s

Learn

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Results of the throw

At t = 2.04 s

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