Intermediate · 12 min
Motion with constant acceleration
Find displacement and final velocity when acceleration stays constant.
Distance during a steady change
For straight-line motion with constant acceleration, displacement is s = u × t + ½ × a × t². Here u is initial velocity, a is acceleration, and t is elapsed time. The first term is the displacement you would get at the initial velocity; the second accounts for the change. Choose a positive direction and use signed values consistently.
Worked example
A cart starts at 2 m/s and accelerates at 1 m/s² for 4 s. s = 2 × 4 + ½ × 1 × 4² = 16 m
When time is not given
The relation v² = u² + 2 × a × s connects initial velocity, final velocity, acceleration, and displacement without time. It also requires constant acceleration. Taking a square root gives a speed magnitude; use the direction of motion to choose a velocity sign. Check that the expression under the square root is nonnegative and that the model matches the situation.
Worked example
A cart starts from rest and accelerates at 2 m/s² through 9 m. v² = 0² + 2 × 2 × 9 = 36 Final speed = 6 m/s
Check a motion model
- Find the final velocity and displacement of a cart starting from rest with acceleration 3 m/s² for 4 s.
- Check the displacement using the area of its velocity–time graph. Explain why the two methods agree.