In a demonstration known as the ballistics cart, a ball is projected vertically upward from a cart moving with constant velocity along the horizontal direction. The ball lands in the catching cup of the cart because both the cart and ball have the same horizontal component of velocity. Now consider a ballistics cart on an incline making an angle with the horizontal as in the figure below. The cart (including wheels) has a mass M and the moment of inertia of each of the four wheels about its axle is mR^2/2.
(a) Using conservation of energy (assuming no friction between cart and axles), and assuming pure rolling motion (no slipping), show that the acceleration of the cart along the incline is

Ax = [M/(M + 2m)] g sin (theta)

(b) Note that the x component of acceleration of the ball released by the cart is g sin theta. Thus, the x component of the cart's acceleration is smaller than that of the ball by the factor M/(M + 2m). Use this fact and kinematic equations to show that the ball overshoots the cart by an amount x, where

change in x = [4m/(M + 2m)] [sin theta/cos^2 theta] Vyi^2/g

and

Vyi is the initial speed of the ball imparted to it by the spring in the cart.

(c) Show that the distance d that the ball travels measured along the incline is

There is something missing in the text with the questions, relevant for question (c):

to make the suggested answer to (c) true, the inventor of the exercise assumed but forgot to mention in the text that in the case of the cart on the slope, the ball is shot up at the moment when the cart is still at rest - that is at the very start of the story.

(a) Suppose the cart started from rest and is now lower by height h from where it started, and it has developed speed v.
Then the amount of potential energy converted into kinetic energy is U = Mgh.
The kinetic energy is
K = Mv^2/2 + 4*(moment of inertia)*(angular velocity)^2/2 =

= Mv^2/2 + 4*(mR^2/2)*(v/R)^2/2 = (M+2m)v^2/2
From K = U we get
(M+2m)v^2 = ...

1. In chronological order, what happens to the kinetic, potential, and total energy of the cart for one half cycle. The half cycle starts just after you have pushed the cart. The half cycle finishes just when the cart stops at the height of its motion. Remember, this is about the cart's mechanical energy.
A. Kinetic goes fro

46) A furniture crate of mass 60.8 kg is at rest on a loading ramp that makes an angle of 25.8 degrees with the horizontal. The coefficient of kinetic friction between the ramp and the crate is .272. What force, (in Newtons) applied parallel to the ramp, is required to push the crate up the incline at a constant speed?
47) A

A BOX IS SLIDING UP AN INCLINE THAT MAKES AN ANGLE OF 16 DEGREES W/ RESPECT TO THE HORIZONTAL. THE COEFFECEINT OF KINETIC FRICTION BETWEEN THE BOX AND THE INCLINE IS 0.18. THE INITIAL SPEED OF THE BOX AT THE BOTTOM OF THE INCLINE IS 3.0 M/S.
HOW FAR DOES THE BOX TRAVEL UP THE INCLINE BEFORE COMING TO REST?

Find the force P necessary to push a 50.6 kg block down an incline of 32.4 degrees with an acceleration of 1.78 m/s^2. P is parallel to the incline and the coefficient of friction between the block and the incline surface is 0.805.
Picture is attached.

A 0.025kg bullet traveling at 305m/s strikes a 1.500 kg ballistics pendulum hanging from a 2.00 m long string and buries itself in the block. How many degrees backward will the block swing? Do not use the law of conservation of momentum.

1) a) Find the angle required to maximize the range down an incline which is pitched at an angle of −θ with respect to the horizontal and the projectile is shot an angle of φ above the incline at a speed of v0 .
b) Determine the distance down the incline at the angle found in a).
c) Plot the trajectory gi

A cart is on an inclined plane. If it is given a push up the ramp and released, then it moves up, reverses direction, and comes back down again. How will the kinetic energy change? How will the gravitational potential energy change? How will the mechanical energy change?

A 2.3-kg cart is rolling across a frictionless, horizontal track toward a 1.5-kg cart that is held initially at rest. The carts are loaded with strong magnets that cause them to attract one another. Thus, the speed of each cart increases. At a sertain instant before the carts collide, the first cart's velocity is +4.5m/s, and th

A small particle slides along a frictionless wire. If the particle's speed at point A is 8.85m/sec how fast is it moving at point B if it must go up a 2m incline?