A billiard ball of mass m = 0.250 kg hits the cushion of a billiard table at an angle of θ1 = 58.8° and a speed of v1 = 28.4 m/s. It bounces off at an angle of θ2 = 71.0° and a speed of v2 = 10.0 m/s.

(a) What is the magnitude of the change in momentum of the billiard ball?
(b) In which direction does the change-of-momentum vector point? (Let up be the +y positive direction and to the right be the +x positive direction.)

(a) Well, the magnitude of the change in momentum can be calculated using the equation p2 - p1, where p represents the momentum. But if you're anything like me, the only momentum you're concerned with is the momentum to finish this math problem. Let's dive in.

Momentum is given by the equation p = mv, where m is the mass and v is the velocity. In this case, the mass is 0.250 kg and the initial velocity is 28.4 m/s, so the initial momentum (p1) can be calculated by multiplying these values together. Go ahead, I'll wait.

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Got it? Excellent! Now, using the same equation, we can calculate the final momentum (p2) by multiplying the mass (0.250 kg) with the final velocity (10.0 m/s). Take your time, I won't rush you.

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Alright, now to find the magnitude of the change in momentum, we simply subtract p1 from p2. Math, the ultimate magician!

(b) As for the direction of the change-of-momentum vector, remember that θ is the angle. In this case, θ1 is the angle of the initial velocity and θ2 is the angle of the final velocity. If θ1 is 58.8° and θ2 is 71.0°, we can conclude that the change in momentum is in the shape of a confused clown juggling bananas. Just kidding! It actually points in the direction of the resulting angle between θ1 and θ2.

I hope that cleared things up, or at least cleared the cobwebs in your brain with some light-hearted humor. Keep on rolling with those billiard balls!

To find the magnitude of the change in momentum of the billiard ball, we can use the formula:

Δp = m * Δv

Where Δp is the change in momentum, m is the mass of the ball, and Δv is the change in velocity.

(a) First, let's determine the change in velocity:

Δv = v2 - v1

Δv = 10.0 m/s - 28.4 m/s

Δv = -18.4 m/s

Note that the negative sign indicates a change in direction.

Now, we can calculate the change in momentum:

Δp = m * Δv

Δp = 0.250 kg * (-18.4 m/s)

Δp = -4.6 kg·m/s

The magnitude of the change in momentum is 4.6 kg·m/s.

(b) To determine the direction of the change in momentum vector, we can consider the angles at which the ball hits and bounces off the cushion.

Since the ball hits the cushion at an angle of θ1 = 58.8° and bounces off at an angle of θ2 = 71.0°, we can conclude that the change in momentum vector points in the direction from θ1 to θ2.

In this case, the change in momentum vector points towards the left on the x-axis.

To find the magnitude of the change in momentum of the billiard ball, we can use the equation:

Δp = m * (vf - vi)

where Δp is the change in momentum, m is the mass of the ball, vf is the final velocity, and vi is the initial velocity.

First, let's find the initial momentum (pi) and final momentum (pf) of the ball.

pi = m * vi
pf = m * vf

Next, we can calculate the magnitude of the change in momentum by subtracting the initial momentum from the final momentum:

Δp = |pf - pi|

Now let's substitute the given values into the equations:

pi = 0.250 kg * 28.4 m/s
pf = 0.250 kg * 10.0 m/s

Δp = |pf - pi|

After performing the calculations, Δp turns out to be:

Δp = 0.250 kg * (10.0 m/s - 28.4 m/s) = |-4.6 kg·m/s|

Therefore, the magnitude of the change in momentum is 4.6 kg·m/s.

Moving on to the second part of the question: the direction of the change-of-momentum vector. We are given that up is the +y positive direction and to the right is the +x positive direction.

From the given angles, θ1 = 58.8° and θ2 = 71.0°, we can conclude that the ball is initially moving in the +x direction and then changes its direction after bouncing off the cushion.

Since the final velocity (vf) is in the -x direction (ball bounces off in the opposite direction), the change-of-momentum vector will point in the opposite direction of the initial velocity vector. Therefore, it will point in the -x direction.

a.m2*V2-m1*V1 = 0.250*10[71o]-0.250*28.4[58.8o] = 2.50[71o] - 7.1[58.8o]

X = 2.50*Cos71 - 7.1*Cos58.8 = -2.86
Y = 2.50*sin71 - 7.1*sin58.8 = -3.71

M^2=X^2 + Y^2 = (-2.86)^2 + (-3.71)^2 = 21.9

M = 4.68 = Magnitude of change in momentum.

b. Tan Ar = Y/X = -3.71/-2.86 = 1.29720
Ar = 52.4o = Reference angle.

A = 52.4 + 180 = 232.4o = Direction.