Question
A cyclist increases her speed uniformly from 5 m/s to 15 m/s in 20 seconds. Calculate her acceleration and the distance traveled in this time.
Quick Answer
Initial velocity u = 5 m/s
Final velocity v = 15 m/s
Time t = 20 s
Acceleration a = (v – u) / t = (15 – 5) / 20 = 0.5 m/s²
Distance s = (u + v) / 2 × t = (5 + 15) / 2 × 20 = 200 m
The cyclist’s acceleration is 0.5 m/s², and the distance travelled in this time is 200 metres.
Understanding the question with detailed explanations
The question is asking us to find the acceleration first. Acceleration is the rate at which velocity changes with time. Since the cyclist’s velocity changes from 5 m/s to 15 m/s in 20 seconds, the increase is steady and uniform. This means the velocity increases by the same amount every second, making it a constant acceleration situation.
The second part of the problem is about finding the distance covered during these 20 seconds. Because the cyclist is accelerating, she does not travel at a single speed but rather starts slower and finishes faster. To calculate the total distance, we use the concept of average velocity during uniform acceleration, which allows us to multiply the average speed by the total time.
A glimpse of the final answer
The calculations show that the acceleration of the cyclist is 0.5 m/s². This means that every second, her speed increases by half a metre per second. Over 20 seconds, this steady increase takes her from 5 m/s up to 15 m/s, which matches the information in the problem.
The distance covered turns out to be 200 metres. This is consistent with expectations because, on average, her speed during the 20 seconds is (5 + 15) / 2 = 10 m/s. If she maintained that average speed for 20 seconds, she would cover 200 metres, which is exactly the result we calculated.
Data
From the problem:
- Initial velocity, u = 5 m/s
- Final velocity, v = 15 m/s
- Time, t = 20 s
What we need:
- Acceleration, a = ?
- Distance travelled, s = ?
All values are already in SI units, so no conversions are required.
Formula
The two relevant equations are:
- a = (v – u) / t
- s = (u + v) / 2 × t
The first equation is the definition of acceleration under uniform conditions, relating change in velocity to time. The second equation is the displacement formula for uniformly accelerated motion, derived from taking the average velocity and multiplying by time.
Both equations are directly applicable because the acceleration is uniform, and the problem provides all the necessary values of u, v, and t.
Solution (solving the problem)
Step 1: Find acceleration.
a = (v – u) / t
a = (15 – 5) / 20
a = 10 / 20
a = 0.5 m/s²
Step 2: Find distance travelled.
s = (u + v) / 2 × t
s = (5 + 15) / 2 × 20
s = 20 / 2 × 20
s = 10 × 20
s = 200 m
Final Answer
The cyclist’s acceleration is 0.5 metres per second squared (0.5 m/s²).
The total distance travelled in the 20 seconds is 200 metres.
Helpful Explanation
This result means that with each passing second, the cyclist steadily increases her speed by half a metre per second, showing how constant effort produces uniform acceleration. By the end of 20 seconds, she has gained an additional 10 m/s in speed, which matches the given information.
A common mistake in this type of problem is forgetting that distance during acceleration is not simply initial velocity multiplied by time. Since the speed changes, the average velocity must be used. An exam tip is to remember that for uniformly accelerated motion, average velocity is always (u + v) / 2. Multiplying that by the time gives the correct displacement every time.