Answer key: Motion Graphs Worksheet: Position, Velocity & Acceleration
For teachers. This page is hidden from search engines and not linked from student pages, but it is not password-protected, so please don’t share the link with students. Student worksheet: Motion Graphs Worksheet: Position, Velocity & Acceleration.
Goal: Connect an object’s motion to its position–time and velocity–time graphs, and use x = v₀t + ½at² and v = v₀ + at to predict where it will be.
Completed results table
| Setting | Time | Position | Velocity |
|---|---|---|---|
| Steady speed (v₀ = 2 m/s, a = 0) | 6 s | 12 m | 2 m/s |
| Speed up (v₀ = 0, a = 2 m/s²) | 6 s | 36 m | 12 m/s |
| Turn around (v₀ = 6 m/s, a = −2 m/s²) | 6 s | 0 m | -6 m/s |
| Turn around, when it stops | 3 s | 9 m | 0 m/s |
| Default (v₀ = 2 m/s, a = 1 m/s²) | 6 s | 30 m | 8 m/s |
Answers
For “Steady speed”, what do the position–time and velocity–time graphs look like? What is the slope of the position–time graph?
The position–time graph is a straight line: the object moves 2 m every second and reaches 12 m at 6 s. Its slope is the velocity, 2 m/s, so the velocity–time graph is a flat line.
For “Speed up”, how far has the object traveled after 6 s? Show the calculation.
36 m. With v₀ = 0 and a = 2 m/s², x = ½at² = ½ × 2 × 6² = 36 m. The position–time graph curves upward because its slope (the velocity) keeps growing, reaching 12 m/s.
In “Turn around” the acceleration is negative from the start. When does the object stop, and where is it at that moment?
At 3 s, 9 m from the start. Setting v = v₀ + at = 6 − 2t equal to 0 gives t = 3 s, and x = 6 × 3 − ½ × 2 × 3² = 9 m. This is the highest point of the position–time graph.
Where is the “Turn around” object at 6 s, and what is its velocity? Explain the sign.
Back at the start (0 m), moving at −6 m/s. After 3 s the negative acceleration makes it speed up in the negative direction, so its velocity is negative: it is heading back toward x = 0.
Predict, then check: with the default settings (v₀ = 2 m/s, a = 1 m/s²), where will the object be at 6 s, and how fast will it go?
At 30 m: x = 2 × 6 + ½ × 1 × 6² = 12 + 18 = 30 m. Its velocity is v = 2 + 1 × 6 = 8 m/s.
Challenge: an object has positive velocity and negative acceleration. Is it speeding up or slowing down? What if both are negative?
Positive velocity with negative acceleration means it is slowing down, because the acceleration points against the motion. If both are negative, it is speeding up in the negative direction. Compare the signs, not just the sign of the acceleration.