Problem 1:

A baseball of mass 200 g is pitched at a speed of 30 m/s towards the batter.  The batted ball's velocity is 60 m/s in the opposite direction.  The ball remained in contact with the bat for 2 ms.  What is the average force applied to the bat?

Problem 2:

A wedge of mass M is moving along a frictionless slope. The slope is fixed to the ground and the slope angle is θ.
The top surface of the wedge M remains horizontal and a cube with mass m is sitting on top of the wedge.  There is also no friction between the cube m and the wedge M.  Starting with Newton's laws,

(a)  find the relative acceleration between m and M, and
(b)  find the magnitude of the normal force N between M and the slope.

Problem 3:

A block of mass 2m is attached to a rigid massless rod of length R and is suspended from a frictionless pivot.  The block is released from rest from position A with the rod extending out horizontally.  When the block swings to reach the bottom of the circle, a bullet of mass m travelling at some velocity v_b strikes it as shown in the figure and lodges in the hole so that the two masses move together as one thereafter.

(a)  Find the block's speed v₀ at the bottom of the circle before being hit by the bullet.
(b)  After the bullet hits the block, the two masses move to the right with a common speed 2v₀.  Find the speed of the bullet v_b just before hitting the block.  Provide your answer in terms of v₀.
(c)  Find the relative change of kinetic energy during this collision.
(d)  Find the tension in the rod immediately after the collision while the two masses are still at the bottom of the circle.  (Draw an appropriate free body diagram and plug v₀ into your answer for the tension).

Problem 4:

An interplanetary spacecraft of mass m and velocity v in a reference frame S with its origin at the sun approaches the planet Jupiter of mass M >> m and velocity V in frame S, as shown in the figure.  With respect to Jupiter, the spacecraft move with velocity v'.  The spacecraft moves around the planet and departs with velocity -v'.  The velocity vector v make an angle of 4 degrees with a line parallel to V.
(a)  What is the speed of the spacecraft in frame S, in terms of V and v after this slingshot maneuver?
(b)  Estimate this speed assuming that the spacecraft was launched from Earth with just enough energy to barely escape the solar system and that Earth's and Jupiter's orbits are circular.

Hint:  Use M_s >> M >> m.

mass of Earth = M_E = 5.98*10²⁴ kg
mass of Jupiter =  318 M_E
mass of Sun = M_S = 1.99*10³⁰ kg
mean radius of Jupiter's orbit = R_J = 7.78*10¹¹ m
mean radius of Earth's orbit = R_E = 1.50*10¹¹ m

Problem 5:

An elevator car of maximum mass m₀ when fully loaded is connected to a counter balance of the same mass by a cable of length L. 
(a)  Derive an expression for the cross sectional area A of the cable such that the yield strength σ_y, (a stress, force/area) of the cable is not exceeded.
(b)  Let ρ be the mass density (mass per unit volume) of the cable and the acceleration due to gravity (9.8 m/s²). 
Assume that m₀ = 8800 kg and the elevator car carries its load up a skyscraper of height 5 km (!). 
Calculate the minimum cross sectional area of the cable without exceeding σ_y for the case of a steel cable. 
Use σ_{y_steel} = 1380 MPa and ρ_steel = 7.7 g/cm³.