Projectile motion: study guide

Key ideas

A projectile moves under gravity alone. Its horizontal and vertical motions are independent: the horizontal velocity stays constant, and the vertical motion has a constant downward acceleration g.

• Split the launch velocity into components: vₓ = v cos θ and v_y = v sin θ.
• The time of flight, when the projectile lands at its launch height, is 2v_y ÷ g, and the maximum height is v_y² ÷ 2g.
• The range is vₓ times the time of flight, R = v² sin 2θ ÷ g. Without air resistance it is largest at 45°.
• For a horizontal launch from height h, the fall time is √(2h ÷ g) and does not depend on the horizontal speed.

Key equations

• vₓ = v cos θ, v_y = v sin θ
• t = 2v_y ÷ g
• H = v_y² ÷ 2g
• R = v² sin 2θ ÷ g
• t = √(2h ÷ g)

Worked example

A ball is kicked at 20 m/s, 30° above the horizontal (g = 9.8 m/s²). v_y = 20 sin 30° = 10 m/s and vₓ = 20 cos 30° = 17.3 m/s. Time of flight = 2 × 10 ÷ 9.8 = 2.04 s. Range = 17.3 × 2.04 = 35.3 m. Maximum height = 10² ÷ (2 × 9.8) = 5.1 m.

Common mistakes

• Using the full launch speed in the vertical equations instead of its vertical component.
• Thinking the horizontal speed decreases, or that a heavier ball falls faster. Without air resistance neither is true.
• Setting the vertical velocity to zero at landing. It is zero only at the highest point.

Practice

Test yourself with practice questions on this topic: Projectile motion