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9. A runner races around a circular track of radius 80 m.

a) Work out the distance the runner covers if the start and finish line are the same. (Use circumference = [tex]2\pi r[/tex].)

b) Find the displacement as he crosses the finish line.

c) If the runner completes one circuit in 60 seconds, what is the average speed?

d) What is the average velocity if the runner races halfway around the field in 30 seconds?

Sagot :

GinnyAnsweridn
Sure, let's tackle each part of the problem step by step.

### a)
The distance that the runner covers if the start and the finish line are the same is the circumference of the circular track. The formula for the circumference of a circle is given by:

[tex]\[ \text{Circumference} = 2 \pi r \][/tex]

where [tex]\( \pi \approx 3.14159 \)[/tex] and [tex]\( r \)[/tex] is the radius of the circle.

Given:
- Radius [tex]\( r = 80 \)[/tex] meters

Thus,
[tex]\[ \text{Circumference} = 2 \times \pi \times 80 \][/tex]
[tex]\[ \text{Circumference} = 2 \times 3.14159 \times 80 \][/tex]
[tex]\[ \text{Circumference} \approx 502.65 \text{ meters} \][/tex]

So, the distance covered by the runner is approximately 502.65 meters.

### b)
The displacement is the straight-line distance from the starting point to the ending point. Since the runner returns to the starting point after one complete circuit, the displacement is:

[tex]\[ \text{Displacement} = 0 \text{ meters} \][/tex]

### c)
The average speed is calculated as the total distance covered divided by the total time taken. The formula for average speed is:

[tex]\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \][/tex]

Given:
- Total distance (Circumference) [tex]\( = 502.65 \)[/tex] meters
- Total time [tex]\( = 60 \)[/tex] seconds

Thus,
[tex]\[ \text{Average Speed} = \frac{502.65}{60} \][/tex]
[tex]\[ \text{Average Speed} \approx 8.38 \text{ meters/second} \][/tex]

### d)
The average velocity is calculated as the total displacement divided by the total time taken. For half the circuit, the formula for average velocity is:

[tex]\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \][/tex]

If the runner races halfway around the track, they cover half of the circumference and the displacement would be the straight-line distance between the starting point and the point directly opposite on the circle.

However, in this context, we look at the displacement along the track making it easier:
- Half of the total distance (Half-Circumference) [tex]\( = \frac{502.65}{2} \approx 251.33 \)[/tex] meters
- Total time for half the circuit [tex]\( = 30 \)[/tex] seconds

Thus,
[tex]\[ \text{Average Velocity} = \frac{251.33}{30} \][/tex]
[tex]\[ \text{Average Velocity} \approx 8.38 \text{ meters/second} \][/tex]

So the average velocity for half the track is approximately 8.38 meters/second.
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