Kinematics

Problem 1:

A cannon ball is shot from ground level towards a target.  Its initial velocity is v0 = 125 m/s at an angle θ = 37 degrees with the horizontal.  Neglect air resistance.
(a)  What are the horizontal and vertical components of the initial velocity?
(b)  What is the maximum height of the cannonball?  How long does it take to reach this height?
(c)  How long does it take to hit the ground?  When it does so, what is its horizontal distance from its starting point?
(d)  What are its height and its horizontal displacement after it has been in the air for 10 s? 
What is its velocity (magnitude and direction) after it has been in the air for 10 s?

Solution:

Problem 2:

Find the minimum rotational frequency (i.e. the number of rotations per second, not the angular frequency) required to rotate a bucket containing 4 liters of water in a circular path in a vertical plane at an arm's length (about 70 cm) without spilling water.

Solution:

Problem 3:

A string with 10 beads (bead #0 to #9) attached to it at equal intervals d0 is dropped from a vertical height h.  (Bead #0 is at height h when the string is released.)  The sound of each bead is heard when it hits the ground.
(a)  Find tn, the time the nth bead hits the ground and evaluate tn/t0.  Does the time interval between sounds increase or decrease a the beads hit the ground?
(b)  Assume we want equally spaced sounds.  If the spacing between bead 0 and bead 1 is d0, what should be the spacing dn between bead n and bead n + 1?

Solution:

Problem 4:

imageA swimmer wants to cross a river, from point A to point B, as shown in the figure.  The distance d1 (from A to C) is 200 m, the distance d2 (from C to B) is 150 m, and the speed vr of the current in the river is 5 km/hour.  Suppose that the swimmer's velocity relative to the water makes an angle of θ = 45o with the line from A to C, as indicated in the figure.
To swim directly from A to B, what speed us, relative to the water, should the swimmer have?

Solution: