Hey, guys. Hopefully, you got a chance to work this one out on your own. Let's go ahead and check it out together. So we have a 4 kilogram box that is moving to the right with an initial speed of 20, and it's going to collide with the spring. The force constant of the spring is 600. What happens is the spring is going to compress. So this box is pushing up against the spring as it collides with it, and then it's going to compress. So the box comes to a stop due to maximum compression when the spring is coiled up a little more.
Now, we want to write our energy conservation equation. We can't solve this using forces because the force that occurs as you're compressing the spring isn't constant; it varies. We have to use energy conservation. Our equation is: K i + U i + W nonconservative = K f + U f . The initial kinetic energy is significant as the box is moving with some speed. There's no initial gravitational potential energy, and since the spring isn't compressed initially, there's no elastic potential energy. There's no work done by non-conservative forces, and the final kinetic energy is zero, so all its kinetic energy has transformed into elastic potential energy.
We write our expressions: 1 2 m v initial 2 = 1 2 k x final 2 . We move k to the other side to get m v initial 2 k = x final 2 . Take the square root to find: x final = m v initial 2 k . Plug in the numbers to get a compression distance of 1.63 meters.
So, after the block has stopped, it has transferred all of its energy and compressed the spring by 1.63 meters. That's the answer. Let me know if you guys have any questions, and I'll see you in the next one.