A 110-kg tackler moving at 2.5 m/s meets head-on (and holds on to) an 82-kg halfback moving at 5.0 m/s. What will be their mutual speed immediately after the collision?
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This is Giancoli Answers with Mr. Dychko. Before the collision, the tackler is moving to the right say, with a velocity of positive 2.5 meters per second and he has a mass of 110 kilograms and this halfback is moving to the left at a velocity of negative 5 meters per second with a mass of only 82 kilograms and after they collide, they are gonna stick together and they are gonna move off with some combined mass equal to m 1 plus m 2 and they will have the same velocity so don't put a subscript v 1 or v 2 prime here, it's just v prime because the first guy and the second guy are both moving with the same speed after the collision because they are stuck together so we'll just call it v prime. And I'm taking a guess that this v prime is going to be to the left since this halfback has twice the speed and more than half of the mass so I know that the product here, the mass times velocity or momentum, in other words, is greater to the left than this momentum to the right; without having to use a calculator or do a calculation, we can just tell by how they, you know, by what proportion are they different. So this is different by a factor of 2, increased velocity, and this is different by something that's more than just a half so there we go; final momentum will be to the left. Anyway, we don't need to know that, that will all come out in the algebra anyway so it doesn't matter what guess you take there. So we have conservation of momentum— that's always going to be true— and we have already said that v 1 prime and v 2 prime are the same since these two guys are stuck together so we'll just it call it v prime which is what I have written in this line here and well since these post-collision velocities are the same, we can factor it out and just multiply that v prime by mass 1 plus mass 2 and then we can solve for that v prime by dividing both sides by m 1 plus m 2 and then switch the sides around and you have v prime is m 1v 1 plus m 2v 2 divided by m 1 plus m 2; you have to be careful with negative signs as you are plugging in numbers here, this v 2 I have taken to be negative since it's to the left so positive is to the right in this picture. And so we have negative 5 meters per second, v 2, velocity of the halfback before the collision times by his mass of 82 kilograms and add to that 110 kilograms times the positive 2.5 meters per second velocity of whatever he is, tackler, divided by the total mass of the two guys and then that makes negative 0.70 meters per second but we are asked for the speed and speed is the magnitude of the velocity and in this case, it's going to be just 0.70 meters per second.
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