Kinematics in One and Two Dimensions
Kinematics in One and Two Dimensions
Kinematics tests description of motion rather than its causes: displacement, velocity and acceleration, the constant-acceleration formulae, motion graphs, calculus links and projectile motion in two dimensions. The mechanics half of the paper expects vector treatment of projectiles and careful separation of horizontal and vertical motion.
Calculus and graphs
v = ds/dt and a = dv/dt = d²s/dt². Going back, velocity is the area under an acceleration–time graph and displacement is the area under a velocity–time graph. On a velocity–time graph the gradient is acceleration, and the total distance travelled is the sum of the magnitudes of the areas, whereas displacement is the signed sum.
Constant acceleration
For constant a: v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t. Choose the equation that contains exactly the three knowns and the one unknown; do not reach for v² = u² + 2as when time is required.
Projectiles
Resolve the initial velocity: uₓ = u cos θ (constant, since horizontal acceleration is zero) and u_y = u sin θ (with vertical acceleration −g). Time of flight to a level surface is 2u sin θ/g, maximum height u²sin²θ/(2g), and range u² sin 2θ/g. Horizontal and vertical motions are independent apart from the shared time.
Worked example
A ball is projected at 25 m s⁻¹ at 30° above horizontal from level ground. Take g = 10 m s⁻². Find the time of flight, range and maximum height. uₓ = 25 cos 30° = 21.65 m s⁻¹ and u_y = 25 sin 30° = 12.5 m s⁻¹. Vertical velocity is zero at the top, so t = 12.5/10 = 1.25 s and the flight time is 2.5 s. Range = 21.65 × 2.5 = 54.1 m. Maximum height = u_y²/(2g) = 156.25/20 = 7.81 m.
Traps that cost marks
- Applying constant-acceleration formulae in the horizontal direction to a projectile.
- Forgetting that vertical velocity at the highest point is zero but the acceleration is not.
- Treating distance and displacement as the same when the particle reverses.
- Sign errors when a particle is projected downwards or from a height.