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Imagine that two vehicles traveling on a frictionless surface crash into each other. Vehicle A is traveling due north, vehicle B is traveling due east, they smash into each other, and B skirts off at an angle exactly 60 degrees south of east while A skirts off at an angle exactly 60 degrees north of east.
The final speeds of A and B, and A's initial northward speed, are not known. All we know is:
1) the initial eastward speed of B (let's call this v0), which is about 64 meters per hour or about 0.018 meters per second
2) that this collision is partially inelastic: not elastic, not superelastic, not completely inelastic
3) the initial and final directions of the velocities of A and B
4) that A has twice the mass of B (if B has mass m, then A has mass 2m)
5) that the system is isolated
Now, how can we determine the range of speeds that A could have had before the collision?
Here was my first thought: using conservation of momentum and conservation of energy, I can find an expression for the ratio of kinetic energy of the system converted to internal energy of the system in the collision, which would be the change in internal energy divided by the initial kinetic energy (which would be the negative of the change of kinetic energy divided by the initial kinetic energy, as the change in potential energy is 0). This ratio must be between 0 and 1, not equal to 0 or 1, as the collision is partially inelastic. Doing all this, I found that the minimum speed that A could have been traveling at would have to be:
[(3/16)^0.5]*v0
where v0 was the initial speed of B (v0 is about 64 meters per hour). This minimum speed is about 27.7 m / hour. However, I couldn't find the maximum speed of A from these constraints, as the ratio is always less than 1 for all speeds of A. So the next thing I thought was, perhaps the maximum speed of A would be the speed at which the maximum amount of kinetic energy is converted to internal energy. This occurs when the initial speed of A equals:
[(3/16)^0.5]*v0*[1+(11/3)^0.5]
At this speed, which is about 80.8 m / hour, the collision converts around 23% of kinetic energy to internal energy.
Am I on the right path? Am I going about this all wrong? Do I need to know something else to find the maximum speed of A before the collision?
Any help would be greatly appreciated. Let me know if I need to explain the problem or my work better (right now, my work is a bit too complicated to post). Thanks!
The final speeds of A and B, and A's initial northward speed, are not known. All we know is:
1) the initial eastward speed of B (let's call this v0), which is about 64 meters per hour or about 0.018 meters per second
2) that this collision is partially inelastic: not elastic, not superelastic, not completely inelastic
3) the initial and final directions of the velocities of A and B
4) that A has twice the mass of B (if B has mass m, then A has mass 2m)
5) that the system is isolated
Now, how can we determine the range of speeds that A could have had before the collision?
Here was my first thought: using conservation of momentum and conservation of energy, I can find an expression for the ratio of kinetic energy of the system converted to internal energy of the system in the collision, which would be the change in internal energy divided by the initial kinetic energy (which would be the negative of the change of kinetic energy divided by the initial kinetic energy, as the change in potential energy is 0). This ratio must be between 0 and 1, not equal to 0 or 1, as the collision is partially inelastic. Doing all this, I found that the minimum speed that A could have been traveling at would have to be:
[(3/16)^0.5]*v0
where v0 was the initial speed of B (v0 is about 64 meters per hour). This minimum speed is about 27.7 m / hour. However, I couldn't find the maximum speed of A from these constraints, as the ratio is always less than 1 for all speeds of A. So the next thing I thought was, perhaps the maximum speed of A would be the speed at which the maximum amount of kinetic energy is converted to internal energy. This occurs when the initial speed of A equals:
[(3/16)^0.5]*v0*[1+(11/3)^0.5]
At this speed, which is about 80.8 m / hour, the collision converts around 23% of kinetic energy to internal energy.
Am I on the right path? Am I going about this all wrong? Do I need to know something else to find the maximum speed of A before the collision?
Any help would be greatly appreciated. Let me know if I need to explain the problem or my work better (right now, my work is a bit too complicated to post). Thanks!
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