A 1300-kg car is to accelerate from rest to a speed of 30.0 m/s in a time of 12.0 s as it climbs a 15.0° hill. Assuming uniform acceleration, what minimum power is needed to accelerate the car in this way?
The minimum power required to accelerate a car with a mass of
Answer to Problem 63SP
Solution:
Explanation of Solution
Given data:
The mass of the car is
The angle of incline of the hill is
The time is
The final velocity is
Formula used:
The expression for change in kinetic energy is written as
Here,
The expression for change in potential energy is written as
Here,
The expression for the work energy theorem is written as
The equation of motion for the final velocity is written as
Here,
The equation of motion for the displacement is written as
Here,
The expression for average power is written as
Here,
Write the expression used for converting power from
Here,
Explanation:
Draw the schematic diagram for the motion of the car:
In the above diagram,
The car accelerates from rest. Therefore, the initial velocity is zero.
Recall the equation of motion for the final velocity of the car:
Substitute
Recall the equation of motion for the displacement of the car:
Substitute
Consider the diagram. The total distance covered by the car is the sine component of the incline. Therefore,
Substitute
The expression for the work energy theorem is written as
Substitute
Substitute
Recall the expression for average power:
Substitute
Recall the expression used for converting power from
Substitute
Conclusion:
The minimum power required to accelerate the car is
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