Projectile motion
Treat the horizontal and vertical motion of a projectile separately, use ΔEp = mgΔh, and explain how air resistance brings a falling body to terminal velocity.
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Before you begin
Projectile motion is motion under gravity near the Earth’s surface. The lessons start with weight and why a projectile moves steadily sideways while accelerating downward, use that to find flight times, ranges and heights, then turn to gravitational potential energy and to how air resistance brings a falling body to terminal velocity.
Be comfortable with: constant-acceleration equations, because the vertical motion of a projectile is motion with constant acceleration.
Learning goals
- Relate weight and gravitational potential energy changes in a uniform gravitational field.
- Analyse projectile motion by separating perpendicular components.
- Explain falling motion with air resistance using forces, energy and terminal velocity.
Syllabus statements covered
- describe and use the concept of weight as the force experienced by a mass in a gravitational field
- describe and explain motion due to a uniform velocity in one direction and a uniform acceleration in a perpendicular direction
- derive, from the definition of work done by a force, the equation ΔEₚ = mgΔh for gravitational potential energy changes in a uniform gravitational field (e.g. near the Earth’s surface)
- recall and use the equation ΔEₚ = mgΔh to solve problems
- describe qualitatively, with reference to forces and energy, the motion of bodies falling in a uniform gravitational field with air resistance, including the phenomenon of terminal velocity
Lessons
Work through them in order.
Core lessons
- Weight and perpendicular motionDescribe weight as a force in a gravitational field, and explain projectile motion as constant horizontal velocity with constant vertical acceleration.
- Projectile motionFind the time of flight, range and maximum height of a projectile that lands at its launch height.
- Unequal-height flights and obstacle clearanceFind the time of flight when a projectile lands above or below its launch point, choose the right root and test whether it clears a wall.
- Gravitational potential energyDerive Δ Eₚ = mgΔ h from the work done by a force, and use it with energy conservation near the Earth's surface.
- Falling with air resistanceExplain, using forces and energy, how air resistance brings a falling body to terminal velocity.
Practise and check
Or choose
Topic reference
Quick reference
| Idea | Relation | Notes |
|---|---|---|
| weight | W = mg | g = 9.81 N kg⁻¹ near the Earth’s surface |
| launch components | uₓ = u cos θ, uy = u sin θ | θ above the horizontal |
| horizontal motion | aₓ = 0, x = uₓt | no air resistance |
| vertical motion | ay = -g, vy = uy-gt, y = uyt-(1/2)gt² | upward positive |
| same launch and landing height | T = 2uy/g, h = uy²/2g, R = (u² sin 2θ)/g | not for unequal heights |
| potential energy | Δ Eₚ = mgΔ h | uniform field |
| energy conservation | v² = u²-2gΔ h | total speeds; gravity the only force doing work |
| terminal velocity | drag = weight | zero acceleration, constant velocity |
Core knowledge to remember
- Weight is the force experienced by a mass in a gravitational field.
- Without air resistance, a projectile has a uniform velocity horizontally and a uniform acceleration g vertically. The two motions are independent, linked only by time.
- At the highest point, only the vertical velocity is zero; the acceleration is still g downward.
- Δ Eₚ = mgΔ h follows from the work done by a force raising a mass in a uniform field, and depends only on the change in height.
- A body falling through air speeds up with decreasing acceleration, because drag grows with speed. At terminal velocity, drag equals weight, the acceleration is zero, and the potential energy lost is all transferred by drag to internal energy.
Common mistakes
- Saying the velocity is zero at the highest point of a projectile’s path.
- Using the same-height results for a projectile that lands higher or lower than its launch point.
- Thinking gravity slows a projectile’s horizontal motion.
- Using the length of a slope instead of the change in height in mgΔ h.
- Saying no forces act, or no energy is transferred, at terminal velocity.