Question

PLEASE ANSWER ALL PARTS

Your 15 year old is about to get their driving permit, and you are concerned about the bumpers on your cars as they are expensive to fix, even after low speed impacts. You decide you want to engineer a true “5 mph” bumper that will slow a collision with a fixed object using a spring attachment, which will avoid damaging the car’s actual bumper. You find out that springs with high spring constant values are very expensive so you decide to first investigate using a smaller and much cheaper setup. You decide to model the situation using a cart on a track with a spring attached to one end and a fixed endstop. You will use this simple setup to first determine the best method to pick the expensive springs for the cars.

track endstop wood block cart with spring attached

The first question deals with the changing momentum of the system:

a. Draw two pictures of the cart: one before the cart hits the end stop and another one after it has bounced off and is no longer in contact with the spring. Label all kinematic quantities and constants in the system. Use the conservation of momentum to write relationship between the motion before and after the collision. What variables in this relationship are measurable with the equipment you have access to?

The remainder of the questions deals with the impulse of the collision:

b. Draw at least four pictures of the cart during the collision with the spring and the end stop including two pictures when the spring is being compressed and two as it expands. Include in each picture the amount of compression in the spring and the direction of the force from the spring on the cart.

c. Write down the relationship of how the force of the spring and the compression of the spring are related in each case. Be sure to name each force something unique (F1, F2, etc.). What, in this relationship, are measurable quantities with the equipment you have access to?

d. Using the four pictures, assume that the time between each picture is equal and that the force in the picture is constant until the next picture. Graph the force of the spring versus time for the duration of the collision.

e. Find the total impulse, by adding together all of the individual areas under the curve in the force versus time graph. Do you expect this estimation to be greater or less than the actual impulse?

f. What are the assumptions made for this model?

g. Using the changing momentum as your ‘theoretical’ calculation, how do you expect the impulse to compare to the changing momentum? Do you expect the time over which the collision occurs to affect how well the impulse agrees with the change in momentum?

Answer 1

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