Certainly! Let's tackle the problem step by step.
### Understanding the Problem
We need to find two things for a child moving on a circular track:
1. Average Speed
2. Average Velocity
Given:
- Radius of the circular track, [tex]\( r = 40 \)[/tex] meters
- Time taken to complete one full revolution, [tex]\( t = 5 \)[/tex] minutes
### (i) Average Speed
Average Speed is defined as the total distance traveled divided by the total time taken.
Step-by-Step Calculation:
1. Find the Circumference of the Circular Track:
- The circumference [tex]\( C \)[/tex] of a circle is given by:
[tex]\[
C = 2 \pi r
\][/tex]
- Substituting [tex]\( r = 40 \)[/tex] meters:
[tex]\[
C = 2 \pi \times 40 = 80 \pi \text{ meters}
\][/tex]
2. Convert Time from Minutes to Seconds:
- Time taken [tex]\( t \)[/tex] = 5 minutes. We need to convert this to seconds.
[tex]\[
t = 5 \times 60 = 300 \text{ seconds}
\][/tex]
3. Calculate Average Speed:
- Average speed [tex]\( v \)[/tex] is given by:
[tex]\[
v = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{C}{t}
\][/tex]
- Substituting the values we found:
[tex]\[
v = \frac{80 \pi}{300} \text{ meters per second}
\][/tex]
- Simplifying this:
[tex]\[
v = \frac{80 \pi}{300} = \frac{80 \pi}{300} \approx 0.84 \text{ meters per second}
\][/tex]
### (ii) Average Velocity
Average Velocity is defined as the total displacement divided by the total time taken. In this scenario, the child returns to the starting point after one full revolution, making the displacement zero.
Step-by-Step Calculation:
1. Find the Total Displacement:
- After one full revolution around the circular track, the displacement is zero because the starting point and ending point are the same.
- Displacement = 0 meters
2. Calculate Average Velocity:
- Since displacement is zero, the average velocity [tex]\( \overline{v} \)[/tex] is:
[tex]\[
\overline{v} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{0}{300} = 0 \text{ meters per second}
\][/tex]
### Summary
1. Average Speed:
[tex]\[
v = \frac{80 \pi}{300} \approx 0.84 \text{ meters per second}
\][/tex]
2. Average Velocity:
[tex]\[
\overline{v} = 0 \text{ meters per second}
\][/tex]
These values provide insight into the child's motion around the circular track. The average speed gives a measure of how fast the child is moving along the track, while the average velocity reflects the overall change in position, which, in this case, is zero as the child ends up where they started.
