Question

2, Figure 1: Figure 2: Consider the model above for an elastic collision used to design bumpers on cars. Car 1 with the bumper (a spring) has an initial speed v and Car 2 is initially at rest. When Car 1 hits Car 2 the bumper compresses some maximum amount, Ax, and then will expand back to its original length. For this problem, let M1- M7 M A) (PS) The system is the two cars and the (ideal) spring bumper. Any friction or air drag on the cars can be neglected. Describe the collision in terms of energy and momentum forms and transformations before, during, and after the cars (and bumper) are in contadt B) (GR) The two figures above show the cars before the collision (Fig. 1) and at the instant of maximum compression of the spring bumper (Fig. 2). Re-draw these figures and draw the system after the collision. On each figure identify all the energies and momenta of the system. Discuss with your TA your answers for Parts A & B before going on. C) (PS) At the instant shown in Fig. 2, with the maximum spring compression, the two cars have the same velocity. Explain why this is true. D) (AR) Solve for an equation (no numbers) for the maximum compression of the spring, ΔΧ, in terms of the mass of the cars, initial speed, and spring constant. Show your work. For the case that M1-M,-103 kg, k-5x 104-ui-2 m, what is x? E) (QC) Explain why/how the equation from Part D makes physical sense if M, k, or v, change F) (Everyone) Bumpers are designed to dissipate the kinetic energy of collisions, not just cushion the collision. To do this you add a device to your bumper so that as the spring compresses there is a force Ffr opposite to the direction of the spring compression. For example, a hydraulic piston might be added to do this

Your design requires that the magnitude of the work done by this device to be equal to ??, where ?? ? ?, from the initial contact to the maximum compression of the bumper spring. What is the magnitude of ?? Important Hint: Δ?, the maximum compression, will change with the energy dissipation device.

Answer 1

Given that To consider the model for an elastic collision used to design bumpers on cars In this the system is the two cars and the spring bumpers . TO describe the collision in terms of energy and momentum forms and transformations before during and after the cars are in contact . Before collision Total momentum = P = (M_1 V_i) + M_2 W = M_1 V_i Total kinetic energy = k . \r\n k = (1/2 M_1 V_i^2 + 0) + (1/2 k (Delta x)^2) Therefore k = 1/2 M_1 V_i^2 Deformation of spring = 0 . During collision Total momentum = p P = M_1 V_c + M_2 V_c p = 2 M V_c Total energy = E E = (1/2 M_1 V^2_c + 1/2 M_2 V^2_c + 1/2 k Delta x^2) After collision perfectly elastic collision Here total momentum = M_1 V i_1 + M_2 V_i_2 (V_