How to Find Change in Velocity: Why It's 7, Not 3

Lesson notes

A short written version of Lesson 8. The study sheet has more practice questions.

Short answer

The change in velocity is 7 m/s, pointing up, not 3 m/s. Take down as the positive direction: v₀ = +5 m/s and vₜ = −2 m/s, so Δv = −2 − 5 = −7 m/s. The minus sign means up.

Key ideas

• Change = final − initial, never initial − final. Money from 3000 to 4000 changes by +1000; coins from 100 to 98 change by −2.
• The change in velocity, Δv, is a vector: an arrow with a size and a direction. Δv = vₜ − v₀, where v₀ is the initial velocity and vₜ is the final velocity.
• If Δv points the same way as v₀, the object speeds up. If Δv points the opposite way and is shorter than v₀, the object slows down.
• If Δv points the opposite way to v₀ and its arrow is longer than v₀, the object turns around.
• On a straight line: choose a positive direction, give v₀ and vₜ their signs, then compute Δv = vₜ − v₀.
• You can calculate Δv from v₀ and vₜ, but they do not cause it. Something outside the object causes it, such as the brake or the gas pedal.

Step by step

1. Choose the positive direction. Take down as positive, the direction of the initial velocity.
2. The ball hits the ground moving down at 5 m/s, so v₀ = +5 m/s.
3. After the bounce it moves up at 2 m/s. Up is opposite to positive, so vₜ = −2 m/s.
4. Compute Δv = vₜ − v₀ = −2 − 5 = −7 m/s.
5. The magnitude is 7 m/s. The minus sign means Δv points up, opposite to the positive direction.
6. So the change in velocity is 7 m/s upward, not 3 m/s.

Common mistakes

• Wrong: computing initial − final. Right: always compute final − initial, or the sign comes out backwards.
• Wrong: subtracting the two speeds, like 5 − 2 = 3. Right: choose a positive direction first, and give each velocity its sign.
• Wrong: dropping the minus sign, like writing 60 km/h for −60 km/h. Right: a + sign can be dropped, but a − sign cannot.
• Wrong: thinking a bigger velocity means a bigger Δv. A bullet train is fast, but at constant velocity its Δv = 0.

Practice

Words in English and Chinese

change in velocity (Δv)

速度变化量

initial velocity

初速度

final velocity

末速度

positive direction

正方向

vector

矢量

scalar

标量

magnitude

大小