The velocity of the system just after the collision using Newton’s second law and the comparison of the answer with the result of Example 11.4.
Answer to Problem 32PQ
The velocity of the system just after the collision using Newton’s second law is
Explanation of Solution
The free-body diagram of the system is shown in figure 1.
The two-train system only moves in
Write the expression for the Newton’s second law in
Here,
Refer to figure 1. The only force acting in
Write the equation for
Here,
Put the above equation in equation (I).
The kinetic friction is proportional to the normal force and the normal force is in turn equal to the weight of the train.
Write the expression for
Here,
Write the equation for
Here,
Put the above equation in equation (III).
Put the above equation in equation (II) and rewrite it for
Initially only the freight train has momentum. Assume that the two trains move together with velocity
Here,
Rewrite the above equation for
Now consider the motion of the system just after the collision to the moment it comes to rest.
Replace
The speed of the two-train system as it comes to rest is zero.
Write the expression for the final speed of the system as it comes to rest.
Here,
Write the constant-acceleration equation of motion.
Here,
Put equations (IV) to (VI) in the above equation and rewrite it for
To find the range of possible initial freight train velocities, the extreme values of the coefficient of kinetic friction must be used.
Write the equation for
Here,
Conclusion:
Given that the value of
Substitute
Substitute
Here,
The system moves in
Here,
Substitute
Substitute
Here,
It is given that the train crosses the red signal at a speed of
Since the train passed the red signal with the speed of
Write the expression for the maximum velocity of the system.
Here,
Substitute
The range of the speed of the two-train system is
Therefore, the velocity of the system just after the collision using Newton’s second law is
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Chapter 11 Solutions
Student Solutions Manual For Katz's Physics For Scientists And Engineers: Foundations And Connections, Volume 1
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