Concept explainers
At an amusement park, there are 200-kg bumper cars A, B, and C that have riders with masses of 40 kg, 60 kg, and 35 kg, respectively. Car A is moving to the right with a velocity vA = 2 m/s and car C has a velocity vB = 1.5 m/s to the left, but car B is initially at rest. The coefficient of restitution between each car is 0.8. Determine the final velocity of each car, after all impacts, assuming (a) cars A and C hit car B at the same time, (b) car A hits car B before car C does.
(a)
Find the final velocity of each car after all impact, assuming car A
Answer to Problem 13.162P
The final velocity of each car after all impact, assuming car A
Explanation of Solution
Given information:
The mass of the bumper car (m) is
The mass of the rider A
The mass of the rider B
The mass of the rider C
The velocity of A
The velocity of C
The coefficient of restitution between each car (e) is 0.8.
Calculation:
Calculate the total mass of car A along with rider
Substitute
Calculate the total mass of the car B along with rider
Substitute
Calculate the total mass of the car C along with rider
Substitute
Assume the velocities towards the right to be positive and the velocities towards the left to be negative.
The velocity will be zero as the car B
The expression for the principle of conservation of momentum to the cars A, B, and C when cars A and C hit the car B at the same time as follows;
Here,
Substitute
Calculate the coefficient of restitution (e) of the impact between the cars A and B using the formula:
Substitute 0.8 for e,
Calculate the coefficient of restitution
Substitute 0.8 for e,
Solve the equations (1) and (2) and (3) to obtain velocities.
Add the equations (2) and (3) to eliminate
Multiply the equation (2) with 260 and subtract it from the equation (1).
Multiply the equations (4) with 500 and add it to the equation (5) to obtain the final velocity of the car C.
Substitute
Substitute
Therefore, the final velocity of each car after all impact, assuming car A
(b)
Find the final velocity of each car after all impact, assuming car A
Answer to Problem 13.162P
The final velocity of each car after all impact, assuming car A
Explanation of Solution
Given information:
The mass of the bumper car (m) is
The mass of the rider A
The mass of the rider B
The mass of the rider C
The velocity of A
The velocity of B
The coefficient of restitution between each car (e) is 0.8.
Calculation:
Calculate the final velocities of the cars when car A hits car B before car C does.
The expression for the principle of conservation of momentum to the first impact between car A and car B as follows:
Substitute
Calculate the coefficient of restitution (e) of the first impact between the cars A and B using the formula:
Substitute 0.8 for e,
Multiply the equations (7) with 240 and add it to the equation (6) to obtain the final velocity of car B.
Substitute
The expression for the principle of conservation of momentum for the second impact between car B and car C as follows:
Here, the final velocity of the car B after the second impact is
Substitute
The expression for the coefficient of restitution
Substitute 0.8 for e,
Multiply the equation (9) with 260 and add it to the equation (8) to obtain the final velocity of car C after the impact.
Substitute
Consider the car A and car B again impact with each other.
The expression for the principle of conservation of momentum to the third impact between the car A and car B as follows;
Here,
Substitute
Calculate the coefficient of restitution (e) of the third impact between the cars A and B using the formula:
Substitute 0.8 for e,
Multiply the equation (11) with 240 and add it to the equation (1) to obtain the final velocity of the car B.
Substitute
Therefore, the final velocity of each car after all impact, assuming car A
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Chapter 13 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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