The Problem:
A roller-skater increases speed from 4 m/s to 10 m/s in 10 seconds at a constant rate. (a) What is her average velocity over this period? (b) For what proportion of the time is she moving at less than her average velocity? (c) For what proportion of the distance is she moving at less than her average velocity?
Douglas Quadling Mechanics1 Exercise1D Q7
A roller-skater increases speed from 4 m/s to 10 ms in 10 seconds at a constant rate. (a) What is her average velocity over this period?
:
Problem Statement:
A roller-skater increases speed from 4 m/s to 10 m/s in 10 seconds at a constant rate.
(a) What is her average velocity over this period?
Solution:
The average velocity is calculated as the sum of the initial and final velocities divided by 2.
Average Velocity = (Initial Velocity + Final Velocity) / 2
Average Velocity = (4 m/s + 10 m/s) / 2
Average Velocity = 14 / 2 = 7 m/s
Answer: The average velocity is 7 m/s.
(b) For what proportion of the time is she moving at less than her average velocity?
Solution:
1. Calculate acceleration:
Acceleration = (Final Velocity – Initial Velocity) / Time
Acceleration = (10 m/s – 4 m/s) / 10 s = 0.6 m/s²
2. Find the time when velocity equals the average velocity (7 m/s):
7 = 4 + 0.6 * t
t = (7 – 4) / 0.6 = 5 seconds
3. Proportion of time below average velocity:
Time below average velocity = 5 seconds
Total time = 10 seconds
Proportion = 5 / 10 = 1/2
Answer: The proportion of time she is moving at less than her average velocity is 1/2.
(c) For what proportion of the distance is she moving at less than her average velocity?
Solution:
1. Calculate total distance:
Distance = Initial Velocity * Time + 0.5 * Acceleration * Time²
Distance = 4 * 10 + 0.5 * 0.6 * 100
Distance = 40 + 30 = 70 meters
2. Calculate distance covered in the first 5 seconds:
Distance = 4 * 5 + 0.5 * 0.6 * 25
Distance = 20 + 7.5 = 27.5 meters
3. Proportion of distance below average velocity:
Proportion = 27.5 / 70 = 11/28
Answer: The proportion of distance she is moving at less than her average velocity is 11/28.
Summary of Answers:
(a) Average velocity: 7 m/s
(b) Proportion of time: 1/2
(c) Proportion of distance: 11/28
