Block slides down a ramp into a spring: impact speed, obtain the maximum compression of the spring.

Block slides down a ramp into a spring: impact speed, obtain the maximum compression of the spring. 🧠 Access full flipped physics courses with video lectures and examples at https://www.zakslabphysics.com/ In this problem, we start with a block at rest on a smooth ramp. There is a spring at the bottom of the ramp, and we are given the distance to the spring. In the first part of the problem, we are asked for the speed of the block when it hits the spring, and we approach the problem using conservation of energy. We set the zero of height at the moment of impact, and we have to compute the initial height using trigonometry since we are given the distance to the spring (hypotenuse) rather than the height above the spring. Then we set up the energy conservation equation, where the initial energy is all gravitational potential energy and the final energy is all kinetic energy because we set the final height to be zero. Thus we obtain the final velocity of the block. Next, we want the maximum compression of the spring, and the key idea here is that the velocity is zero at maximum compression. We label a compression distance d, and we keep the same zero for the y coordinate, so the final height of the mass is negative when the spring is compressed. Again we have to use trigonometry to express the final height of the mass in terms of d, taking care to say d is a positive number and including a minus sign to indicate the final height is negative (and thus the gravitational potential energy is negative). We set up the energy conservation equation, including initial gravitational potential energy, final gravitational potential energy, and final spring energy. When we simplify the energy conservation equation, we get a quadratic equation in terms of spring compression, d. We use a CAS to solve the quadratic equation and we pick the positive solution for d. Finally, we show how to set up the quadratic formula by hand to solve for d. We put all the coefficients in the right places, simplify the square root and take the positive solution, and we obtain the maximum compression of the spring.