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 = ...

Explain why the work required to pull a dynamics cart up an incline, in the absence of friction, should be the same as the work required to lift the cart vertically through the vertical displacement it experiences in the process.

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

Show ALL your work, including the equations used to solve the problems.
A cookie jar is moving up a 40º incline. At a point 55 cm from the bottom of the incline (measured along the incline), it has a speed of 1.4 m/s. The coefficient of kinetic friction between jar and incline is 0.15.
a) How much farther up the incline

A spherical shell starts from rest and rolls down an incline. The coefficient of static friction between the shell and incline is (Us=.2). What is the maximum angle of incline such that the shell will roll without slippage?

An employee is pushing ten empty shopping carts, lined up in a straight line. The acceleration of the carts is .050 m/s^2. The ground is level, and each cart has a mass of 26 kg.
What is the net force acting on any one of the carts and assuming friction is negligible.
what is the force exrted by the fifth cart on the sixth

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?

Please see the attached file.
A block of mass m = 2.00 kg is released from rest at h = 0.400 m from the surface of a table, at the top of a theta = 40.0 degrees incline as shown below. The frictionless incline is fixed on a table of height H = 2.00 m.
(a) Determine the acceleration of the block as it slides down the incli

Please see the attachment for full description and figure
The spring shown in Figure P11.54 is compressed 40 cm and used to launch a 100 kg physics student. The track is frictionless until it starts up the incline. The student's coefficient of kinetic friction on the 30° incline is 0.15.
(a) What is the student's speed j