Concept explainers
In a women’s 100-m race, accelerating uniformly, Laura takes 2.00 s and Healan 3.00 s to attain their maximum speeds, which they each maintain for the rest of the race. They cross the finish line simultaneously, both setting a world record of 10.4 s. (a) What is the acceleration of each sprinter? (b) What are their respective maximum speeds? (c) Which sprinter is ahead at the 6.00-s mark, and by how much? (d) What is the maximum distance by which Healan is behind Laura, and at what time does that occur?
(a)
The acceleration of each sprinter.
Answer to Problem 2.83CP
The acceleration of Laura and Healan are
Explanation of Solution
The time taken by Laura and Healan to attain their maximum speeds are
Write the expression for the distance covered by Laura
Here,
Substitute
Write the expression for the distance covered by Healan
Here,
Substitute
Conclusion:
Therefore, the acceleration of Healan and
(b)
The maximum speeds of Laura and Healan.
Answer to Problem 2.83CP
The maximum speeds of Laura and Healan are
Explanation of Solution
Write the formula to calculate the maximum speed for Laura
Here,
Substitute
Therefore, the maximum speed for Laura is
Write the formula to calculate the maximum for Healan
Here,
Conclusion:
Substitute
Therefore, the maximum speed for Healan is
(c)
The leading sprinter at
Answer to Problem 2.83CP
The sprinter is Laura is ahead of Healan by
Explanation of Solution
Write the formula to calculate the distance for Laura at
Write the formula to calculate the distance for Healan at
Write the expression for the difference of distance travelled by Laura and Healan
Substitute
Conclusion:
Substitute
The positive sign shows that Laura is ahead of Healan.
Therefore, the sprinter Laura is ahead of Healan by
(d)
The maximum distance by which Healan is behind Laura and time at which maximum distance occurs.
Answer to Problem 2.83CP
The maximum distance by which Healan is behind Laura is
Explanation of Solution
Maximum distance between runners occurs during the time interval when Laura has stopped accelerating but Healan is still accelerating. The separation is maximum, when both of them have the same velocity.
Equate both velocities at a time
Here,
Substitute
Write the formula to calculate the distance for Laura at
Here,
Write the formula to calculate the distance for Healan from equation (II)
Write the expression for the maximum distance by which Healan is behind Laura
Substitute
Conclusion:
Substitute
Therefore, the maximum distance by which Healan is behind Laura is
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