Concept explainers
(a)
The time interval during which the bicycle is ahead of the car.
(a)
Answer to Problem 38P
The time interval during which the bicycle is ahead of the car is
Explanation of Solution
Write the
Here,
Rewrite the above equation for
Use the above equation to write the expression for the time taken by the bicycle to reach its maximum speed.
Here,
It is given that the acceleration of the car is less than the acceleration of the bicycle. This implies the car cannot catch the bicycle until the time bicycle is at its maximum speed and coasting and it will be greater than the time taken by the bicycle to reach its maximum speed.
Write the equation for the total displacement of the bicycle.
Here,
It is given that the car starts from rest so that its initial velocity will be zero.
Write the equation for the displacement of the car during the time taken by the bicycle to reach the car.
Here,
When the car catches the bicycle, the two displacements will be equal.
Write the condition when the car to catches the bicycle.
Conclusion:
Substitute
Substitute
Substitute
Put equations (V) and (VI) in equation (IV).
Write the quadratic formula to solve the equation
Equation (VII) is a quadratic equation in
Since
Therefore, the time interval during which the bicycle is ahead of the car is
(b)
The maximum distance by which the bicycle leads the car.
(b)
Answer to Problem 38P
The maximum distance by which the bicycle leads the car is
Explanation of Solution
The distance by which bicycle will lead the car will increase as long as the bicycle moves faster than the car or when the speed of the car becomes equal to the maximum speed of the bicycle.
Write the equation for the elapsed time when the bicycle’s lead ceases to increase.
Here,
Write the equation for the lead of the bicycle.
Here, the subscript
Put equations (V) and (VI) in the above equation.
Conclusion:
Substitute
Substitute
Therefore, the maximum distance by which the bicycle leads the car is
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Chapter 2 Solutions
Principles of Physics: A Calculus-Based Text
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