A bored wizard wandered into a gym and noticed a volleyball lying on the floor.

Question

A bored wizard wandered into a gym and noticed a volleyball lying on the floor. On a whim, raised his wand, muttered some words, and the volleyball shot straight up into the air. The volleyball has a circumference of 66 cm and a mass of 270 gram. Due tothe magical impulse that the wizard applied to the ball, its initial upward velocity was 15.5 m/s with a spin velocity of 480 RPM.The top of the ball rotates toward the Northwhich we’ll also call the X-axis.After casting his initial spell that set the volleyball in motion, the wizard wanders off forgetting about the volleyball which is now subject to just normal kinematic/dynamic behavior.Ignoring air friction, answer the following questions:

1.What impulse(Ft) was required to accelerate the ball upward at 15.5 m/s? Express the answer in units of N·s

2.What rotation impulse (t)was required to spin the ball at 480 RPM? Express the answer in units of m·N·s.(Although this assumption is not correct, for this assignment treat the ball as a solid uniform sphere.)A rotational impulse is similar to a translational impulse in that it causes a change in momentum a translational impulse causes a change in translational momentum, p, and a rotational impulse causes a change in rotational momentum, L.

3.How far upward will the ball travel?

4.How high should the gym ceiling be so that the ball would just almost touch the ceiling?(Assume there are no obstructions between the ball and the ceiling.)

5.What is the velocity vector,(vx, vy)and velocity magnitude, |v|, of a spot on the top side of the ball at the top of its trajectory? Give the vector in the x- y plane where x is defined as positive point to the North and y is positive in upward direction.

6.What is the velocity vector,(vx, vy), and velocity magnitude, |v|, of a spot on the top side of the ball at the instant before it hits the gym floor?(Assume that the ball did not touch the ceiling.) Give the vector in the x-y plane where x is defined as positive point to the North and y is positive in upward direction.

7.Assuming the ball does not slip when it impacts the ground, what direction will the ball move when it lands on the floor? Give the direction as an angle where North is 0° and West is 90°.(You only need to give the direction; you do not need to give the speed.)

8.How long was the ball in the air?

9.How many revolutions did the spinning ball make during its first 2.0 seconds in the air?

Explanation / Answer

1. impulse, FdT = Change in momentum = 0.270*15.5 = 4.185 Ns
2. Angular impulse, TdT = Change in angular momentum = Iw = 2mr^2w/5 = 2*0.270*[0.66/2pi]^2 *480*2pi/60*5 = 0.059929 Nms
3. mgh = 0.5*mv^2
h = 0.5v^2/g = 0.5*15.5^2/9.8 = 12.25m
4. Hieght of the gym cieling = h = 12.25 m
5. At the top of trajectory, Vy = 0, Vx = r*w = [0.66/2pi]*[480*2pi/60] = 0.66*480/60 = 5.28 m/s
v = 5.28 i
|v| = 5.28 m/s

6. Same as 5
7. Assuming elastic collision with the floor
Velocity of the bottom most point just before collision = -5.8i - 15.5 j
After collision, velocity = 5.8i + 15.5 j [bottom most point]
hence the angle mustb e arctan(15.5/5.8) = 69.48 deg

8. v = u - gt t = u/g = 15.5/9.8 = 1.581 s
9. s = ut - 0.5 at^2
s = 480*2pi*2/60 = 100.48 rad
s = 16 rev

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