Complete Question:
In the "before" part of Fig. 9-60, car A (mass 1100 kg) is stopped at a traffic light when it is rear-ended by car B (mass 1400 kg). Both cars then slide with locked wheels until the frictional force from the slick road (with a low ?k of 0.15) stops them, at distances dA = 6.1 m and dB = 4.4 m. What are the speeds of (a) car A and (b) car B at the start of the sliding, just after the collision? (c) Assuming that linear momentum is conserved during the collision, find the speed of car B just before the collision.
Answer:
a) Speed of car A at the start of sliding = 4.23 m/s
b) speed of car B at the start of sliding = 3.957 m/s
c) Speed of car B before the collision = 7.28 m/s
Explanation:
NB: The figure is not provided but all the parameters needed to solve the question have been given.
Let the frictional force acting on car A,
............(1)
Since frictional force is a type of force, we are safe to say
.......(2)
Equating (1) and (2)
![ma = \mu mg\\a = \mu g\\\mu = 0.15\\a = 0.15 * 9.8 = 1.47 m/s^{2}](https://tex.z-dn.net/?f=ma%20%3D%20%5Cmu%20mg%5C%5Ca%20%3D%20%5Cmu%20g%5C%5C%5Cmu%20%3D%200.15%5C%5Ca%20%3D%200.15%20%2A%209.8%20%3D%201.47%20m%2Fs%5E%7B2%7D)
a) Speed of A at the start of the sliding
![d_{A} = 6.1 m\\Speed of A at the start of sliding, v_{A} = \sqrt{2ad_{A} }\\ v_{A} = \sqrt{2*1.47*6.1 } \\v_{A} = \sqrt{17.934 } \\v_{A} = 4.23 m/s](https://tex.z-dn.net/?f=d_%7BA%7D%20%3D%206.1%20m%5C%5CSpeed%20of%20A%20at%20the%20start%20of%20sliding%2C%20v_%7BA%7D%20%3D%20%5Csqrt%7B2ad_%7BA%7D%20%7D%5C%5C%20v_%7BA%7D%20%3D%20%5Csqrt%7B2%2A1.47%2A6.1%20%7D%20%5C%5Cv_%7BA%7D%20%3D%20%5Csqrt%7B17.934%20%7D%20%5C%5Cv_%7BA%7D%20%3D%204.23%20m%2Fs)
b) Speed of B at the start of the sliding
![d_{A} = 4.4 m\\Speed of A at the start of sliding, v_{B} = \sqrt{2ad_{B} }\\ v_{B} = \sqrt{2*1.47*4.4 } \\v_{B} = \sqrt{12.936 } \\v_{B} = 3.957 m/s](https://tex.z-dn.net/?f=d_%7BA%7D%20%3D%204.4%20m%5C%5CSpeed%20of%20A%20at%20the%20start%20of%20sliding%2C%20v_%7BB%7D%20%3D%20%5Csqrt%7B2ad_%7BB%7D%20%7D%5C%5C%20v_%7BB%7D%20%3D%20%5Csqrt%7B2%2A1.47%2A4.4%20%7D%20%5C%5Cv_%7BB%7D%20%3D%20%5Csqrt%7B12.936%20%7D%20%5C%5Cv_%7BB%7D%20%3D%203.957%20m%2Fs)
Let the speed of car B before collision = ![v_{B1}](https://tex.z-dn.net/?f=v_%7BB1%7D)
Momentum of car B before collision = ![m_{B} v_{B1}](https://tex.z-dn.net/?f=m_%7BB%7D%20v_%7BB1%7D)
Momentum after collision = ![m_{A} v_{A} + m_{B} v_{B2}](https://tex.z-dn.net/?f=m_%7BA%7D%20v_%7BA%7D%20%2B%20m_%7BB%7D%20v_%7BB2%7D)
Applying the law of conservation of momentum:
![m_{A} = 1100 kg\\m_{B} = 1400 kg](https://tex.z-dn.net/?f=m_%7BA%7D%20%3D%201100%20kg%5C%5Cm_%7BB%7D%20%3D%201400%20kg)
![(1400*v_{B1} ) = (1100 * 4.23) + ( 1400 * 3.957)\\(1400*v_{B1} ) = 10192.8\\v_{B1} = 10192.8/1400\\v_{B1 = 7.28 m/s](https://tex.z-dn.net/?f=%281400%2Av_%7BB1%7D%20%29%20%3D%20%281100%20%2A%204.23%29%20%2B%20%28%201400%20%2A%203.957%29%5C%5C%281400%2Av_%7BB1%7D%20%29%20%3D%2010192.8%5C%5Cv_%7BB1%7D%20%3D%2010192.8%2F1400%5C%5Cv_%7BB1%20%3D%207.28%20m%2Fs)