A 30 g dart traveling horizontally hits and sticks in the back of a 500 g toy car, causing the car to roll forward at 1.4 m/s. What was the speed of the dart?

Video transcript

Hey, everyone. So this problem is dealing with conservation of momentum. Let's see what it's asking us. A bullet with a mass of 80 g is fired horizontally into a block of wood with a mass of 800 g causing the block to slide forward with a velocity of 2.5 m per second. The bullet gets embedded in this wood, meaning that it's an an elastic collision determine the initial speed of a bullet before it collides with the block of wood, which is initially at rest. Our multiple choice answers here are a 27. m per second. B 14.2 m per second. C 33.8 m per second or D 22.4 m per second. OK. So this is a pretty straightforward conservation of momentum equation where we can recall that our initial momentum is equal to our final momentum and momentum as give is given, excuse me, as P equals mass multiplied by philosophy. So we're going to take each part of the initial system. So we have the bullet and the block initially and then we'll have the bullet in the block and the block finally after the collision, so we'll have mass of the bullet multiplied by the speed. The initial speed of the bullet plus mass of the, we'll call it, call it W for the wooden block plus seed of the wooden block initially is equal to mass of the bullet multiplied by speed of the bullet after the collision, plus the mass of the wooden block multiplied by speed of the wooden block after the collision. And so when we look at each of these terms, one by one, we are solving for the initial speed of the bullet. So that's that term there. So mass of the bullet we're given is 80 g. I'm gonna rewrite that as 0. kg. To keep us in standard units, mass of the wooden block is 800 g or 0.8 kg. The block slides forward with a velocity of 2.5 m per second, but the bullet is actually embedded in the block. And so this those speeds are the same, the speed of the block and the speed of the bullet after the collision are going to be the same. They're both gonna be 2.5 m per second. And our initial speed of the uh block is zero, it's initially at rest. And so we have everything we need to solve for this um mass of or sorry, we have everything we need to solve for this initial speed of the bullet. So we're just going to plug in all of these known values mass of the block, 0.8 kg multiplied by that speed of the bullet. Sorry, the mass of the bullet multiplied by the speed of the bullet. Initially that we're looking for mass of the wooden block is 0.8 kg multiplied by our initial speed which is zero. So that term goes away equals mass of the bullet. Again, 0.8 kg multiplied by 2.5 m per second plus. So that's mass of the bullet multiplied by the speed of the bullet and the block together at the end plus the mass of the block. So that's 0.8 kg multiplied by that same speed. 2.5 m per second. Yeah. Can then solve for this B B I, we plug all of this into our calculator and we are left with 27.5 m per second. And so that is the correct answer for this problem and aligns with answer choice. A it's all I've got for this one. We'll see you in the next video.