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. This practice uses idealized vector models and separates horizontal and vertical motion, without claiming a target paper structure.
Calculus and graphs
v = ds/dt and a = dv/dt = d²s/dt². Going back, the change in velocity is the signed 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
- Giving a projectile a nonzero horizontal acceleration in the ideal no-air-resistance model; the constant-acceleration equations remain valid horizontally with a = 0.
- 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.