Inelastic Collisions in One Dimension

[beatbama85]

Here's my problem:

Car A (mass 970 kg) is stopped at a traffic light when it is rear-ended by car B (mass 1600 kg). Both cars then slide with locked wheels until the frictional force from the slick road (with a low mk of 0.23) stops them, at distances dA = 5.8 m and dB = 3.6 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.

The name of the section that this problem corresponds to is the title of this thread. I know about the conservation of momentum (ma(va1) + mb(vb1) = ma(va2) + mb(vb2), where a and b are the cars and 1 and 2 is initial and final) and the equation for the center-of-mass velocity ((pa1 + pa2)/(ma + mb)), but I don't know what to do with the friction. Should I find the frictional forces and integrate them to get the final momentums?

 

[beatbama85]

Nevermind, I can't believe I didn't think of this earlier. I just realized that, after the collision, the kinetic energy transfers completely to thermal energy, so I was able to find va2 and vb2 from K = 1/2v^2. Then, applying the equation for conservation of momentum, I was able to find vb1.

 

[thepatientmental]

Great job identifying the relevant equations and concepts for this problem! To solve for the speeds of the cars at the start of sliding, you can use the conservation of momentum equation and the given information about the distances traveled. Since the cars are sliding with locked wheels, the frictional force will be equal to the maximum static friction force (fs = μsN). You can then use this force to solve for the initial momentums of the cars and then use those values to solve for the initial speeds.

To find the speed of car B just before the collision, you can use the same equation for conservation of momentum, but instead of using the distances traveled, you can use the initial and final velocities of car A (which is stopped at the traffic light) and the final velocity of car B (which is the same as its initial velocity just before the collision). This will give you one equation with one unknown, which you can then solve for the initial velocity of car B.

As for the frictional forces, you do not need to integrate them to find the final momentums. The frictional force will only affect the motion of the cars while they are sliding, and once they come to a stop, the frictional force will no longer be present and the cars will no longer be experiencing any acceleration. This means that you can solve for the final momentums directly using the given information about the distances traveled.

Overall, it seems like you have a good understanding of the concepts and equations needed to solve this problem. Just remember to pay attention to the given information and use it appropriately in your calculations. Good luck!