Two 0.50-kg carts are pushed toward each other from starting positions at either end of a 6.0-m low-friction track. Each cart is pushed with a force of 3.0 N, and that force is exerted for a distance of 1.0 m. (a) What is the work done on the two-cart system? (b) What is the change in kinetic energy of the system? (c) What is the kinetic energy of the center of mass of the system?

Two 0.50-kg carts, one red and one green, sit about half a meter apart on a low-friction track. You push on the red one with a constant force of 2.0 N for 0.15 m and then remove your hand. The cart moves 0.35 m on the track and then strikes the green cart. (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 red cart? (c) By what amount does your force change the kinetic energy of the center of mass of the system?

In a carnival game, the player throws a ball at a haystack. For a typical throw, the ball leaves the hay with a speed exactly one-half of the entry speed. (a) If the frictional force exerted by the hay is a constant 6.0 N and the haystack is 1.2 m thick, derive an expression for the typical entry speed as a function of the inertia of the ball. Assume horizontal motion only, and ignore any effects due to gravity. (b) What is the typical entry speed if the ball has an inertia of a 0.50 kg?

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?

Suppose a jogger has to exert a force of 25 N against air resistance to maintain a velocity of +5.0 m/s. At what rate is the jogger expending energy?

A 2.34-kg cart on a long, level, low-friction track is heading for a small electric fan at 0.23 m/s. The fan, which was initially off, is turned on. As the fan speeds up, the magnitude of the force it exerts on the cart is given by $a t ^ { 2 }$, where $a = 0.0200 \mathrm { N } / \mathrm { s } ^ { 2 }$. What is the speed of the cart 3.5 s after the fan is turned on? After how many seconds is the cart’s velocity zero?

A farmer is trying to haul three 500-kg trailers connected together with identical ropes that will break if their tension exceeds 2000 N. The first trailer is connected to his tractor with a rigid hitch. (a) If the tractor has to overcome a frictional force of 900 N per trailer to start them moving, what is the minimum force the tractor must exert on the first trailer to get them all rolling if the ropes are taut? (b) What is the maximum force the tractor can exert without breaking any rope?

You want to hang an object from the ceiling of an elevator that has a maximum acceleration of $4.0 \mathrm { m } / \mathrm { s } ^ { 2 }$. (a) If you hang the object with fishing line that supports 45 N of force, what is the maximum inertia the object can have if the line is not to break? (b) What combination of slowing down, speeding up, going up, and going down of the elevator causes the greatest force to be exerted on the fishing line?

A red 10-kg cart is connected to a 20-kg cart by a relaxed spring of spring constant 60 N/m. The 20-kg cart is resting against another 10-kg cart, this one blue. All are on a low-friction track. You push the red 10-kg cart to the right, in the direction of the 20-kg cart, with a constant force of 10 N. (a) What is the acceleration of the center of mass of the three-cart system? (b) What is the acceleration of each cart the instant you begin to push? (c) What is the vector sum of the forces on each cart when the spring is compressed 100 mm?

A 1500-kg truck and a 1000-kg car are parked with their rear bumpers nearly touching each other in a level parking lot. Both vehicles have their brakes off so that they are free to roll. A man sitting on the rear bumper of the truck exerts a constant horizontal force on the rear bumper of the car with his feet, and the car accelerates at $1.2 \mathrm { m } / \mathrm { s } ^ { 2 }$. (a) What are the magnitude and direction of the acceleration of the center of mass of the car-truck system? (b) What vector sum of forces (magnitude and direction) is exerted on each vehicle? (c) What are the magnitude and direction of the acceleration of the truck? Ignore any friction between the tires and the parking lot surface.

Decide if the differential equation $\ddot{\theta}+\sin\theta=1$ is conservative or nonconservative and describe its phase portrait.

How did Gregory I increase the political power of the pope?

Two blocks are hung by a string draped over a pulley, a 1.0-kg block on the left and a 3.0-kg block on the right. The two blocks start out at rest and at the same height. (a) What is the change in the gravitational potential energy of the system of blocks and Earth when the 3.0-kg block has dropped 0.53 m? (b) What is the change in the kinetic energy of the system between release and this instant? (c) What is the velocity of the 1.0-kg block at this instant?

A 68.5 kg astronaut is doing a repair in space on the orbiting space station. She throws a 2.25 kg tool away from her at 3.20 m/s relative to the space station. What will be the change in her speed as a result of this throw?