1.4 Analytical Solutions to Problems Involving Motion

Kinematic Equations for Constant Acceleration

When acceleration is constant, four equations link position, velocity, acceleration, and time: v = v₀ + at, x = x₀ + v₀t + ½at², v² = v₀² + 2a(x − x₀), and x = x₀ + ½(v₀ + v)t. Pick the equation that already contains the variable you don't know, or that omits the one you don't need, to solve efficiently.

Setting Up the Problem

Always define a positive direction, list known/unknown variables, and check whether acceleration is truly constant before applying kinematic equations. Free fall problems use a = −g (9.8 m/s²) if up is positive. Careful sign conventions prevent errors in velocity and displacement outcomes.

Multi-Stage Motion

Some problems involve multiple phases with different accelerations (e.g., speeding up then slowing down). Solve each stage separately, using the final velocity/position of one stage as the initial condition for the next. Total displacement or time is the sum across all stages.

Key terms

v = v₀ + at

Kinematic equation relating final velocity to initial velocity, acceleration, and time (no displacement).

x = x₀ + v₀t + ½at²

Kinematic equation for position over time using initial velocity and acceleration (no final velocity needed).

v² = v₀² + 2a(x − x₀)

Kinematic equation relating velocities, acceleration, and displacement (no time needed).

Free fall acceleration

Constant acceleration due to gravity, magnitude 9.8 m/s², directed downward.

Multi-stage motion strategy

Break motion into segments with constant acceleration, solve each separately, and use end conditions of one stage as start conditions of the next.

Try an AP-style question

A cart moving at 5 m/s accelerates at 2 m/s² for 3 s. What is its final velocity?

1. 8 m/s
2. 6 m/s
3. 15 m/s
4. 11 m/s