kinematics 1d 2d

Kinematics is the study of motion without considering the forces causing it. In Cambridge A-Level Mechanics, we analyze motion in one dimension (1D - straight line) and two dimensions (2D - plane).

Fundamental Quantities

Displacement (s): Vector quantity measuring change in position from origin (units: m). Not the same as distance traveled.

Velocity (v): Rate of change of displacement; v = ds/dt (units: m s⁻¹). Velocity is a vector with magnitude and direction.

Acceleration (a): Rate of change of velocity; a = dv/dt = d²s/dt² (units: m s⁻²). Can be positive (speeding up in positive direction) or negative (slowing down or speeding up in negative direction).

Essential Equations (SUVAT)

For constant acceleration only:

• v = u + at
• s = ut + ½at²
• s = ½(u + v)t
• v² = u² + 2as
• s = vt - ½at²

Where: u = initial velocity, v = final velocity, a = acceleration, s = displacement, t = time

Mnemonic: "Sally Uses Very Awkward Tables" helps remember the five variables.

2D Motion

In 2D kinematics, we resolve motion into perpendicular components (typically horizontal x and vertical y). Each component follows 1D kinematic equations independently.

Position vector: r = xi + yj

Velocity vector: v = (dx/dt)i + (dy/dt)j

Acceleration vector: a = (d²x/dt²)i + (d²y/dt²)j

For projectile motion under gravity: horizontal acceleration = 0, vertical acceleration = -g (≈ -9.8 m s⁻²).

Understanding 1D Motion

Imagine a train journey between two stations. The displacement is the straight-line distance from start to finish (perhaps 5 km north), while the distance traveled includes all curves and turns. If the train reverses, displacement could be zero even though distance traveled is significant.

Velocity vs. Speed: A car traveling around a roundabout at constant speed has changing velocity because direction changes continuously. Speed is scalar (50 km/h), velocity is vector (50 km/h eastward).

2D Motion Applications

Projectile Motion - Football Kick: When a goalkeeper kicks a ball at angle θ to horizontal with speed u:

• Horizontal component: u_x = u cos θ (remains constant - no air resistance)
• Vertical component: u_y = u sin θ (changes due to gravity)

The ball follows a parabolic trajectory. Maximum height occurs when vertical velocity = 0. Range depends on both components.

River Crossing Problem: A swimmer aims perpendicular to a river bank but water current carries them diagonally. Resultant velocity = vector sum of swimming velocity and current velocity. To reach directly opposite point, swimmer must aim upstream at calculated angle.

Calculus Connection

Velocity-time graphs: area under curve = displacement (integration). Gradient = acceleration (differentiation).

Real-world insight: Tesla autopilot systems constantly calculate position by integrating acceleration data from sensors - exactly like solving kinematics problems!

Elevator Motion: Starting from rest, accelerating upward (positive a), constant velocity (a = 0), then decelerating (negative a) before stopping demonstrates all three motion phases clearly.