Concept explainers
(a)
Whether the rocket designed to be used to sample the local atmosphere for pollution, achieves its goal of reaching 20 km.
(a)
Answer to Problem 79P
The rocket reaches a height of
Explanation of Solution
Given:
The initial velocity of the rocket
The magnitude of its upward acceleration
Time during which the rocket accelerates upwards
The height the rocket should reach
Formula used:
The rocket accelerates upwards for a time
Assume a sign convention where the
The distance travelled by the rocket during this time period is given by the following expression:
At the end of the time interval
After a time
Here,
The total height travelled by the rocket is the sum of the two distances
Calculation:
Calculate the vertical distance
Using equation (2), calculate the value of the rocket’s speed
After the engines are switched off, the rocket decelerates due to the action of the acceleration of free fall. When it reaches the maximum point in its trajectory, its final velocity becomes zero.
Substitute
Therefore,
Calculate the maximum vertical distance travelled by the rocket by substituting the calculated values of
Conclusion:
The rocket was designed to sample air at a height of
For the rocket to reach
(b)
The total time the rocket is in air.
(b)
Answer to Problem 79P
The rocket is in air for a total time of
Explanation of Solution
Given:
The velocity of the rocket when engine is switched off
The total vertical distance travelled by the rocket.
Time during which the rocket accelerates upwards
Velocity at the point of maximum height
Formula used:
The time taken by the rocket to reach its maximum height after the engine is switched off is calculated using the expression:
When the rocket reaches the point of maximum height its velocity becomes zero and it starts to fall down.
The time taken by the rocket to fall is given by the following expression:
The negative sign shows that the displacement is made in the downward direction.
The total time the rocket is in air is the sum of (i) the time taken by it to accelerate upwards(ii) time taken to reach the maximum height after the engine is switched off and (iii) time taken to fall to the ground from the point of maximum height.
Therefore,
Calculation:
Substitute the given values of variables in equation (5) and calculate the time
Substitute the values of variables in equation (6) and calculate the time
Substitute the values of
Conclusion:
Thus, the rocket is in air for a total time of
(c)
To determine the speed of the rocket just before it hits the ground.
(c)
Answer to Problem 79P
The speed of the rocket just before it hits the ground is found to be
Explanation of Solution
Given:
The vertical distance the rocket falls from the point of maximum height.
Velocity at the point of maximum height
Time taken by the rocket to fall to the ground
Formula used:
The speed of the rocket when it just reaches the ground is calculated using the following expression:
Calculation:
Substitute the values of the variables in equation (8) and calculate the rocket’s speed when it hits the ground.
The negative sign shows that its velocity is directed downwards along the −y direction.
Conclusion:
Thus, the speed of the rocket just before it hits the ground is found to be
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Chapter 2 Solutions
Physics For Scientists And Engineers Student Solutions Manual, Vol. 1
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