A 6.0-kg bowling ball is held 10 mm above a mattress and then released from rest and allowed to fall, sinking into the mattress. Model the mattress as a single spring with spring constant k = 500 N/m. (a) Calculate the ball’s kinetic energy, the gravitational potential energy of the system (ball, spring, and Earth), and the elastic potential energy at the following compression's: 0, 0.050 m, 0.10 m, 0.15 m, 0.20 m. (b) What is the maximum compression of the mattress? Ignore the inertia of the spring.

Because the soles of your shoes have cleats, you can exert a forward force of 100 N even on slippery ice. A 10-kg picnic cooler is at rest on a frozen pond, and you want to get it to shore, up the bank, and stuck in the snow. From past experience, you know that, in order to move up the bank and stick, the cooler needs to be moving at 3.0 m/s at the instant it starts up the bank. Standing some distance from the cooler and not wanting to walk to it, you decide to throw 1.0-kg snowballs at it to get it moving. If each snowball smashes into the cooler in a totally inelastic collision, what minimum number of snowballs must you throw?

A shopping cart that has an inertia of 12 kg when empty is loaded with 38 kg of groceries. A child pushing the loaded cart loses control, and the cart rolls into a concrete lamppost, which sustains no damage. The cart is moving at 2.0 m/s when it strikes the lamppost, and the groceries all crash forward on impact, moving the center of mass forward by 0.10 m before everything comes to a halt. What are (a) the average force exerted by the post on the cart-groceries system, (b) the work done by the post on the system, and (c) the change in kinetic energy of the system’s center of mass?

You devise a wind-up car powered by a spring for trips to the grocery store. The car has an inertia of 500 kg and is 4.2 m long. It should be able to accelerate from rest to 20 m/s at least 50 times before the spring needs winding. The spring runs the length of the car, and a full winding compresses it to half this length. In order to meet the acceleration requirement, what must the value of the spring constant be?

While you are driving on a highway, a friend in another car passes you. You accelerate at a constant rate to catch up. When you do catch up, are your two cars going the same speed? Use a graph to support your answer.

Set up, but do not evaluate, the following optimization problem.
You are building five identical pens adjacent to each other with a total area of $1000 m ^2$. What dimensions should you use to minimize the amount of fencing?

Solve each inequality. $r-4\leq3$

Candace has $1.55 in quarters, dimes, and nickels. She has the same number of quarters as dimes. The dimes are worth$0.30 in all. How many coins are dimes or quarters? Explain.

Two children on ice skates pull toward each other on a rope held taut between them. The inertia of one child is 30 kg, and the inertia of the other is 25 kg. (a) If at one instant the 30-kg child is accelerating at $1.0 \mathrm { m } / \mathrm { s } ^ { 2 }$ to the left, what is the acceleration of the 25-kg child at that instant? (b) What happens to each acceleration if the 25-kg child stops pulling and just hangs on?

You have a 2.0-m chain lying on the floor alongside ten 0.10-kg cubical blocks, each 0.20 m on a side. Each block is resting on the floor, and the inertia of the chain is 1.0 kg. Which process requires less work: lifting one end of the chain so that the chain hangs vertically with the bottom link just touching the floor, or stacking the blocks in a column ten blocks tall?

A thin film of glycerin (n = 1.473) of thickness 524 nm with air on both sides is illuminated with white light at near normal incidence. What wavelengths will be strongly reflected in the range 300 nm to 700 nm?

Two identical 0.50-kg carts, each 0.10 m long, are at rest on a low-friction track and are connected by a spring that is initially at its relaxed length of 0.50 m and is of negligible inertia. You give the cart on the left a push to the right (that is, toward the other cart), exerting a constant 5.0-N force. You stop pushing at the instant when the cart has moved 0.40 m. At this instant, the relative velocity of the two carts is zero and the spring is compressed to a length of 0.30 m. A locking mechanism keeps the spring compressed, and the two carts continue moving to the right. (a) What is the work done by you on the two-cart system? (b) How far does the system’s center of mass move while you are pushing the left cart? (c) By what amount do you change the system’s kinetic energy?

Determine whether the following statements are true and give an explanation or counterexample. if f'(c)=0, then f has a local maximum or minimum at c.

In a chart, list each early battle of the American Revolution, its outcome, and why it was important.