Equal changes in equal times

The idea

For an object moving in a straight line in one direction, acceleration is constant if the magnitude of its velocity changes by equal amounts in equal intervals of time, for all choices of intervals.

Counting the speedometer

A speedometer is read every second. In one car the reading rises 3, 6, 9, 12. In another it rises 1, 4, 9, 16.

Which car do you think has a steady (constant) acceleration?

• The one whose reading rises by the same amount each second
• The one whose reading becomes larger
• The one with the larger last reading

Constant or not?

Look at how much the velocity changes in each second, then sort.

Same change, every interval

In the steady cases the velocity changed by the same amount in every second, whether it was rising or falling. In the others the amount changed from second to second. The book asks for this to hold for all choices of time intervals, not just for one.

For an object moving in a straight line in one direction, the acceleration is constant if the magnitude of its velocity increases or decreases by equal amounts in equal intervals of time, for all possible choices of intervals.

Notes

Constant acceleration: the magnitude of velocity changes by equal amounts in equal intervals of time, for all choices of intervals.

Check yourself

The velocity of a car reads 2, 4, 6, 8 m s⁻¹ at successive seconds. Its acceleration is…

The velocity of a car reads 0, 1, 4, 9 m s⁻¹ at successive seconds. Its acceleration is…

Acceleration is constant if the magnitude of velocity…

A car has a velocity of 20 m s⁻¹ at 0 s, 16 m s⁻¹ at 1 s and 12 m s⁻¹ at 2 s, and it keeps falling by the same amount each second. What is its velocity at 4 s?

Answer: 4 m s⁻¹

20, 16, 12, 8, 4: at 4 s the velocity is 4 m s⁻¹.

• not constant. The velocity rises by 2 m s⁻¹ in every second: equal amounts in equal intervals, so the acceleration is constant.
• zero. The velocity changes, so the acceleration is not zero.
• constant — correct. Yes!
• not constant — correct. Yes!
• constant. The velocity rises by 1, 3, 5 m s⁻¹ in successive seconds. These amounts are unequal.
• zero. The velocity changes, so the acceleration is not zero.
• is large. A large velocity does not decide the acceleration. What matters is how the velocity changes with time.
• changes by equal amounts in equal intervals of time — correct. Yes!
• changes by different amounts in equal intervals. That is the case where the acceleration is not constant.