For motion in a straight line with constant acceleration a, use the equations of motion. u is the initial velocity, v the final velocity, s the displacement and t the time.
These four equations are used whenever the acceleration is constant, such as in free fall (a = g = 9.8 m s⁻²).
Projectile motion
A projectile moves under gravity only. Treat the horizontal and vertical motions separately: the horizontal velocity is constant (no horizontal force) and the vertical motion has acceleration g downward. If a projectile is launched at speed u at angle θ above the horizontal:
The range is greatest at θ = 45° when air resistance is ignored.
Try itA projectile
Choose the launch speed and the angle. Find the time of flight, the range and the greatest height. (g = 9.8 m/s²)
Launched at 20 m/s at 30°: the path is a parabola with range about 35 m.
Newton's laws, momentum and collisions
Law
Statement
First
An object stays at rest or moves at constant velocity unless a resultant force acts
Second
The resultant force equals the rate of change of momentum: F = Δ(mv)/Δt, or F = ma for constant mass
Third
Forces between two bodies are equal and opposite and act on different bodies
In any collision or explosion in an isolated system, the total momentum is conserved. In an elastic collision kinetic energy is also conserved.
Worked exampleA collision
A 2 kg trolley moving at 3 m/s hits a stationary 1 kg trolley and they stick together. Find their common velocity.
Momentum before: 2 × 3 + 1 × 0 = 6 kg m/s.
After the collision the trolleys move together with total mass 3 kg.
Answer: The trolleys move together at 2 m/s. (Kinetic energy is lost: it is an inelastic collision.)
Circular motion and gravitation
An object in circular motion at constant speed has an acceleration towards the centre (centripetal). ω is the angular velocity in radians per second.Newton's law of gravitation: G = 6.67 × 10⁻¹¹ N m² kg⁻². The Earth’s field strength g = 9.8 N/kg at the surface.
Work, energy, power and equilibrium
Work done is the energy transferred. Energy is conserved. For equilibrium the resultant force is zero and the total moment about any point is zero.
Key terms
Projectile
An object moving under gravity alone.
Momentum
Mass multiplied by velocity.
Centripetal force
The inward resultant force that keeps an object in circular motion.
Impulse
Force multiplied by the time it acts; equal to the change in momentum.
Summary
Use v = u + at, s = ut + ½at², v² = u² + 2as for constant acceleration.
Resolve projectile motion into constant horizontal velocity and vertical acceleration g.
Momentum is conserved in collisions; F = Δp/Δt.
Circular motion needs a centripetal force F = mv²/r; gravity follows an inverse-square law.
