An athlete completes one round of a circular track of diameter 100 m in 20 s. What will be the displacements after 1 minute and 10 s, respectively ?
0 m, 100 m
Calculating Athlete Displacement on a Circular Track
This problem involves calculating the displacement of an athlete running on a circular track. Displacement is the shortest straight-line distance between the initial and final positions of an object. It is a vector quantity, meaning it has both magnitude and direction.
Understanding the Circular Track and Motion
- The track is circular with a diameter of \(\text{100 m}\).
- The radius of the track is half of the diameter, which is \(\text{100 m} / 2 = \text{50 m}\).
- The athlete completes one full round of the track in \(\text{20 s}\). This is the time period for one complete cycle.
Displacement After 1 Minute
First, let's calculate the displacement after \(\text{1 minute}\).
- Total time = \(\text{1 minute} = \text{60 seconds}\).
- Time taken for one round = \(\text{20 seconds}\).
- Number of rounds completed in \(\text{60 seconds} = \frac{\text{Total time}}{\text{Time per round}} = \frac{\text{60 s}}{\text{20 s}} = 3 \text{ rounds}\).
After completing 3 full rounds on the circular track, the athlete returns to their starting position. Displacement is the change in position. Since the athlete is back at the start, the final position is the same as the initial position.
Therefore, the displacement after \(\text{1 minute}\) is \(\text{0 m}\).
Displacement After 1 Minute and 10 Seconds
Next, let's calculate the displacement after \(\text{1 minute and 10 seconds}\).
- Total time = \(\text{1 minute and 10 seconds} = \text{60 seconds} + \text{10 seconds} = \text{70 seconds}\).
- Time taken for one round = \(\text{20 seconds}\).
- Number of rounds completed in \(\text{70 seconds} = \frac{\text{Total time}}{\text{Time per round}} = \frac{\text{70 s}}{\text{20 s}} = 3.5 \text{ rounds}\).
This means the athlete completes 3 full rounds and then an additional half round (\(\text{0.5}\) rounds).
- The first 3 full rounds bring the athlete back to the starting point, resulting in \(\text{0}\) displacement from the start after these rounds.
- The additional \(\text{0.5}\) rounds means the athlete runs exactly halfway around the circular track from the starting point.
If the athlete starts at point A and runs half a round, they will end up at point B, which is directly opposite to A on the circular track. The straight-line distance between two diametrically opposite points on a circle is equal to the diameter of the circle.
The diameter of the track is given as \(\text{100 m}\).
Therefore, the displacement after \(\text{1 minute and 10 seconds}\) (3.5 rounds) is equal to the diameter of the track, which is \(\text{100 m}\).
Summary of Displacements
| Time | Number of Rounds | Final Position | Displacement |
|---|---|---|---|
| 1 minute (60 s) | 3 | Back at start | 0 m |
| 1 minute 10 seconds (70 s) | 3.5 | Diametrically opposite to start | 100 m |
The displacement after \(\text{1 minute}\) is \(\text{0 m}\).
The displacement after \(\text{1 minute and 10 seconds}\) is \(\text{100 m}\).
Revision Table: Key Concepts for Circular Motion Displacement
| Concept | Description | Application in this Problem |
|---|---|---|
| Displacement | Shortest straight-line distance from initial to final position. Vector quantity. | Calculated based on the net change in position. |
| Circular Track | Motion along a circle. | The path of the athlete is circular. |
| One Round | Completing a full circle. | Takes 20 s. Brings athlete back to start. |
| Full Rounds > Displacement | After completing any integer number of full rounds, displacement is zero. | 3 full rounds result in 0 m displacement. |
| Half Round > Displacement | After completing an odd multiple of half rounds (0.5, 1.5, 2.5...), displacement is equal to the diameter. | 3.5 rounds means half a round from the start, resulting in 100 m displacement. |
Additional Information: Displacement vs. Distance
It is important to distinguish between displacement and distance covered.
- Distance: The total length of the path covered by the athlete. It is a scalar quantity.
- Displacement: The change in position from the start to the end point. It is a vector quantity.
In this problem:
- After \(\text{1 minute (60 s)}\), the athlete completes 3 rounds. The distance covered would be \(3 \times (\text{Circumference}) = 3 \times \pi \times \text{100 m} = 300\pi \text{ m}\). However, the displacement is \(\text{0 m}\).
- After \(\text{1 minute and 10 seconds (70 s)}\), the athlete completes 3.5 rounds. The distance covered would be \(3.5 \times (\text{Circumference}) = 3.5 \times \pi \times \text{100 m} = 350\pi \text{ m}\). However, the displacement is \(\text{100 m}\) (the diameter).
Understanding this difference is key when solving problems involving motion.
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