Motion in One Dimension

Constant Acceleration

speeding up from rest

When a cart speeds up steadily from rest, its speed grows with time and its distance grows with time squared, so equal seconds make growing gaps.

Example

When a cart speeds up steadily from rest, its speed grows with time and its distance grows with time squared, so equal seconds make growing gaps.

highlighted = computed this step

Read the motion

A cart starts from rest and speeds up at 2 metres per second squared for 4 seconds. Starting from rest means the first speed is zero.

a=2 m/s2,t=4 sa = 2\ \text{m}/\text{s}^{2}, \quad t = 4\ \text{s}
A cart at rest, about to accelerateA cart at the starting line with an acceleration arrow pointing in the direction it will speed up.carta

Speed after the time

With no starting speed, velocity is acceleration times time: 2 times 4 is 8 metres per second. The seconds-squared underneath cancels one second, leaving metres per second.

v=a t=2 m/s2 ⋅ 4 s=8 m/sv = a\,t = 2\ \text{m}/\text{s}^{2} \,\cdot\, 4\ \text{s} = \hl{8}\ \text{m}/\text{s}

The distance formula from rest

Distance from rest is one half the acceleration times the time squared. The one half is there because the speed builds up evenly from zero.

x=12 a t2x = \tfrac{1}{2}\,a\,t^{2}

Substitute the numbers

Put in 2 for the acceleration and 4 for the time. The time squared is 16.

x=12 ⋅ 2 m/s2 ⋅ 42=12 ⋅ 2 ⋅ 16 mx = \tfrac{1}{2} \,\cdot\, 2\ \text{m}/\text{s}^{2} \,\cdot\, 4^{2} = \tfrac{1}{2} \,\cdot\, 2 \,\cdot\, 16\ \text{m}

Compute the distance

That gives 16 metres. Notice the snapshots at each second spread further apart: equal times, growing gaps.

x=16 mx = \hl{16}\ \text{m}
Equal time steps land at growing gapsSnapshots at each whole second spread further and further apart and the velocity arrows grow, because the cart keeps speeding up.t=1t=2t=3cartv
mechanics Starting from rest with a clean acceleration keeps every step exact, and the growing gaps between equal-time snapshots are the whole point.