Newtonian mechanics

Uniformly accelerated motion

Constant acceleration makes velocity change uniformly with time and position vary quadratically with time.

When a vehicle accelerates steadily, its velocity increases by the same amount each second. The v–t graph is straight, while x–t curves.

v = v₀ + at, x = x₀ + v₀t + ½at²

Definition: Quantities and model

Acceleration a is constant; v₀ and x₀ are velocity and position at t = 0. The sign of a depends on the chosen direction. These equations apply to straight motion with constant acceleration.

Use the simulation to observe motion and read quantitative values as time evolves.

Reading the model

Constant acceleration makes velocity change uniformly with time and position vary quadratically with time.

Example: Worked example

A vehicle has v₀ = 2 m/s and a = 1 m/s². After 3 s, v = 2 + 1×3 = 5 m/s; displacement is 2×3 + ½×1×9 = 10.5 m.

Solution

A vehicle has v₀ = 2 m/s and a = 1 m/s². After 3 s, v = 2 + 1×3 = 5 m/s; displacement is 2×3 + ½×1×9 = 10.5 m.

With constant acceleration, velocity changes by equal amounts in equal time intervals. The equations $v=v_0+at$ and $s=v_0t+ rac12at^2$ use signed displacement, so choose a positive direction before substituting. A car moving at 5 m/s and accelerating at 1.5 m/s² for 4 s reaches 11 m/s and travels 32 m during that interval.

Example: Calculating a car’s motion

A car starts at 2 m/s and accelerates at 1 m/s² for 5 s. Find final speed and displacement.

Solution

v=2+1×5=7 m/s; s=2×5+½×1×25=22.5 m.

If motion is opposite the positive direction, $v_0$ or $a$ may be negative; do not take absolute values before substituting. The time-free relation $v^2-v_0^2=2as$ is useful when displacement is known but time is not. Check the direction and units of the result.

Quick check

Which quantity or statement is correct for this motion?

Which statement correctly describes the motion in this topic?

References

Young, Freedman (2019). University Physics
Halliday, Resnick, Walker (2014). Fundamentals of Physics