Momentum, Collisions and Explosions Legacy Problem #31 Guided Solution

An effective problem solver reads a problem carefully and develops a mental picture of what's going on, perhaps even diagramming the situation. An effective problem solver identifies the known and the unknown information, perhaps even listing the known quantities up on the diagram. An effective problem solver, before ever using the calculator, plots out a strategy to get from known to unknown information. Here we have a collision, which I would term a true dimensional collision, in which the two objects, instead of heading in the same direction or the directly opposite direction, are heading at right angles to one another. One of the vehicles is a Ford Explorer driven by Robin Banks, and we know its mass to be 2080 kilograms, and it's moving north at 32.6 meters per second. I draw an arrow heading up on the page of paper, and I list next to the arrow the mass of the Ford Explorer and the velocity of the Ford Explorer. The other vehicle is a garbage truck whose mass to state is 1,800 kilograms, and it's moving east at 12.4 meters per second. What I now do is draw an arrow heading to the east, and I label it 12.4 meters per second with an M equal 1840 kilograms. We want to find the post-collision speed of these two vehicles if they entangle together as a result of the collision and slide as a single object across the road. So the way I'm going to approach this is I'm going to find the momentum of the garbage truck before the collision. It's an east momentum, and I'm going to find the momentum of the Ford Explorer before the collision. It's a north momentum. These momentums are vectors, they have direction, and we can add them together to find the total momentum if we apply rules of vector addition. Those rules for right angle vectors involve the use of the Pythagorean theorem. So I can take the momentum of the garbage truck squared plus the momentum of the Explorer squared, and that's equal to the total momentum squared. I can solve for total momentum. That's the momentum before the collision, and since momentum gets conserved, that's the total momentum of the system after the collision. Now all of this post-collision momentum is shared by both objects. They're moving as though they're a single object, and I can determine the mass of this single object, so to speak, by summing the individual masses of the truck and the Explorer. So here we go. I can employ that strategy to find the post-collision speed and direction of the two objects entangled together. So for the garbage truck, I take its mass and its velocity, and I multiply them together to get the momentum of the garbage truck. It comes out to be 228160 kilograms per meter per second. I do the same thing for the Explorer, 2080 kilograms times 32.6 meters per second gives me a momentum for the Explorer, 67808 kilograms per meter per second. Since these vectors are at right angles to one another, adding them to determine the whole involves the Pythagorean theorem. So I go truck squared plus Explorer squared equals square of the total, and by the time I solve for this, I get 2323822.92 kilograms per meter per second as the total momentum. Now this total momentum is shared by both objects moving together as though one, and the mass of these objects is 20480 kilograms. So I'm going to take the total momentum, and I'm going to divide it by the total mass. So using the equation P equals mv, and using total values, I'll be able to calculate the velocity of these two objects after the collision. It comes out to be 11.62 meters per second, and I'll round that to 11.6 for three significant digits. Now to determine the direction of these two objects, I have to go back to my diagram. I have the Explorer heading north and the truck heading east, and I found their momentum. It's momentums that add together, and so to find their post-collision direction, what I have to do is find the direction of the total momentum vector. The direction of the vector that forms the hypotenuse of a right triangle that has as its sides the momentum of the truck and the momentum of the Explorer before the collision. So that involves drawing a hypotenuse, adding them together, drawing a right triangle, getting the hypotenuse of that right triangle, and finding the angle that it makes with east. To do that, I have to use a tangent function, and I have to say the tangent of theta is the side opposite theta, which would be the northerly side of that right triangle. It's 67808 units of momentum. Divided by the side adjacent, which is the truck's momentum, it's 228160. So if I go theta equals the inverse tangent of 67808 over 228160, I will get as an angle value 16.58 degrees, or rounded to 16.6 degrees, and that's north of east.