Lesson 9 of 10

What Is Acceleration? Why It's 180, Not 20

加速度

Today's question

You gain 30 points in 120 days, your friend gains 10 points in 10 days. Who improved faster?

Lesson notes

A short written version of Lesson 9. The study sheet has more practice questions.

Short answer

Your friend improved faster. Divide each change by its time: you gain 30 ÷ 120 = 0.25 points per day, and your friend gains 10 ÷ 10 = 1 point per day.

Key ideas

Change tells you how much something changed. Rate of change = change ÷ time interval, and it tells you how fast it changed.
A large change is not always a fast change. A large change over a very long time gives a small rate of change.
Acceleration is the rate of change of velocity: a = Δv / Δt. The unit is m/s², because Δv is in m/s and Δt is in s.
The name can mislead. Acceleration can describe velocity increasing or decreasing.
a = Δv / Δt gives the average acceleration. Example: +2, +10 and +1 m/s in 3 s gives a = 13 m/s ÷ 3 s ≈ 4.3 m/s².
Acceleration is a vector. It points the same way as Δv.

Step by step

The change alone is not enough. +30 points is bigger than +10 points, but it took 120 days, not 10 days.
To see how fast each person improved, divide the change by the time interval. Here the unit time is one day.
You: 30 points ÷ 120 days = 0.25 points per day.
Your friend: 10 points ÷ 10 days = 1 point per day.
1 > 0.25, so your friend improved faster. A bigger change does not mean a faster change.
Acceleration works the same way for velocity: a = Δv / Δt, the change in velocity divided by the time.

Common mistakes

Wrong: a large change in velocity always means a large acceleration. Right: also look at the time. A large Δv over a very long time gives a small a.
Wrong: high speed means large acceleration. Right: a maglev train moves fast but usually at constant velocity, so its a = 0.
Wrong: giving the ball a positive velocity after it bounces off the wall. Right: take inward as positive, so v₁ = −8 m/s and Δv = −8 − 10 = −18 m/s.
Wrong: a negative acceleration is a small acceleration. Right: the sign of a only shows direction. To compare magnitudes, compare the numbers only.

Practice

Question 1. A ball hits a wall horizontally at 10 m/s. After 0.1 s of contact with the wall, it bounces back along the same straight line at 8 m/s. What is the ball's acceleration during the collision? ( )
A. 20 m/s², perpendicular to the wall, inward
B. 20 m/s², perpendicular to the wall, outward
C. 180 m/s², perpendicular to the wall, inward
D. 180 m/s², perpendicular to the wall, outward

Show the answer

D. Take inward (toward the wall) as positive. v₀ = +10 m/s and v₁ = −8 m/s. Δv = −8 − 10 = −18 m/s. a = Δv / Δt = −18 / 0.1 = −180 m/s². The magnitude is 180 m/s². The sign of a is negative, so it is opposite to the positive direction (inward). It points outward.

Question 2. An object's initial velocity is 6 m/s. After 10 s, the magnitude of its velocity is 20 m/s. The magnitude of its acceleration could be ( ) (more than one may be correct)
A. 0.8 m/s²
B. 1.4 m/s²
C. 2.0 m/s²
D. 2.6 m/s²

Show the answer

B and D. Take the direction of the initial velocity as positive, so v₀ = +6 m/s. Only the magnitude of the final velocity is given, so there are two cases. ① Same direction: v = +20 m/s, Δv = 20 − 6 = 14 m/s, a = 14 / 10 = 1.4 m/s² (B). ② Opposite direction: v = −20 m/s, Δv = −20 − 6 = −26 m/s, a = −26 / 10 = −2.6 m/s². The magnitude is 2.6 (D).

Words in English and Chinese

acceleration (a): 加速度
rate of change: 变化率
change in velocity (Δv): 速度变化量
average acceleration: 平均加速度
vector: 矢量
positive direction: 正方向
magnitude: 大小

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