Speeding Up or Slowing Down? Check a and v

Lesson notes

A short written version of Lesson 10. The study sheet has more practice questions.

Short answer

You cannot tell from how a changes. Look at the directions of a and v together. Same direction means speeding up, and opposite directions mean slowing down.

Key ideas

• v = v₀ + Δv. So v depends on v₀ (initial velocity) and Δv. Δv depends on a and the time t.
• a does not depend on v or Δv. a decides the direction of Δv, but not its magnitude, because that also depends on t.
• Speeding up or slowing down: a and v in the same direction means speeding up. Opposite directions mean slowing down.
• Neither v alone nor a alone is enough. It does not matter whether a gets bigger or smaller.
• If a = 0, the object is not necessarily at rest. It may be in uniform straight-line motion.
• If a and v are opposite, the object slows down until v = 0. a is still there, so it turns around and speeds up the other way.

Step by step

1. Do not look at whether a gets bigger or smaller. Look at the directions of a and v.
2. Case 1: a and v point the same way, say to the right. The car speeds up, even while a gets smaller.
3. Example with made-up numbers: v = 10, 15, 17, 18, 18.3 m/s at t = 1 to 5 s.
4. Each second v grows by 5, 2, 1, then 0.3 m/s. v still increases, just more and more slowly. When a reaches 0, v stays constant.
5. Case 2: a and v point opposite ways. The car slows down, whether a gets bigger or smaller.
6. So "a keeps getting smaller" alone cannot tell you. You need the directions of both a and v.

Common mistakes

• Wrong: a gets smaller, so v gets smaller. Right: if a and v point the same way, v still increases while a decreases.
• Wrong: a = 0 means the object is at rest. Right: a = 0 means v does not change. The object may move at constant velocity.
• Wrong: positive a means speeding up and negative a means slowing down. Right: the sign of a shows only direction. You also need the direction of v.
• Wrong: −4 m/s² is smaller than 3 m/s². Right: compare magnitudes by the numbers only. 4 > 3, so −4 m/s² is the greater acceleration.

Practice

Words in English and Chinese

depends on

取决于

change in velocity (Δv)

速度变化量

initial velocity

初速度

speeding up / slowing down

加速 / 减速

same direction / opposite direction

同向 / 反向

constant-acceleration linear motion

匀变速直线运动

displacement

位移

net force

合外力