A billiard ball of mass m = 0.150 kg hits the cushion of a billiard table at an angle of θ1 = 60.0° at a speed of v1 = 2.50 m/s. It bounces off at an angle of θ2 = 47.0° and a speed of v2 = 2.20 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? (Take the x-axis along the cushion and specify your answer in degrees.)
Thank you for Any and All Help!!
To find the magnitude of the change in momentum of the billiard ball, we first need to calculate the initial and final momenta.
a) The initial momentum (p1) can be calculated using the formula:
p1 = m * v1
where
m = mass of the billiard ball = 0.150 kg
v1 = initial velocity = 2.50 m/s
Substituting the given values, we get:
p1 = 0.150 kg * 2.50 m/s
p1 = 0.375 kg⋅m/s
Similarly, the final momentum (p2) can be calculated using the formula:
p2 = m * v2
where
v2 = final velocity = 2.20 m/s
Substituting the given values, we get:
p2 = 0.150 kg * 2.20 m/s
p2 = 0.330 kg⋅m/s
The magnitude of the change in momentum is given by:
Δp = |p2 - p1|
Substituting the calculated values, we get:
Δp = |0.330 kg⋅m/s - 0.375 kg⋅m/s|
Δp = |-0.045 kg⋅m/s|
Δp = 0.045 kg⋅m/s
Therefore, the magnitude of the change in momentum of the billiard ball is 0.045 kg⋅m/s.
b) To determine the direction of the change in momentum vector, we need to consider the angles of impact and rebound.
Since the initial angle of impact (θ1) is 60.0° and the final angle of rebound (θ2) is 47.0°, the change in momentum vector points in the direction between these two angles.
The change in momentum vector points in the direction halfway between θ1 and θ2. Therefore, the direction of the change of momentum vector is given by:
(θ1 + θ2) / 2
Substituting the given values, we get:
(60.0° + 47.0°) / 2
= 107.0° / 2
= 53.5°
Therefore, the direction of the change of momentum vector is 53.5 degrees with respect to the x-axis, along the cushion of the billiard table.
To find the magnitude of the change in momentum of the billiard ball, we need to calculate the final momentum (p2) and initial momentum (p1), and then find the difference between them (Δp = p2 - p1).
The momentum of an object is given by the product of its mass and velocity:
p = mv
a) Calculation of initial momentum (p1):
The initial velocity components can be calculated using the given speed and angle:
v1x = v1 * cos(θ1)
v1y = v1 * sin(θ1)
Using the given mass (m = 0.150 kg), the initial momentum can be found:
p1 = m * v1
b) Calculation of final momentum (p2):
Similar to above, we can calculate the final velocity components:
v2x = v2 * cos(θ2)
v2y = v2 * sin(θ2)
And then find the final momentum using the given mass:
p2 = m * v2
c) Calculation of the change in momentum (Δp):
Now we can find the change in momentum by subtracting the initial momentum from the final momentum:
Δp = p2 - p1
To determine the direction of the change in momentum, we can calculate the angle it makes with the x-axis.
d) Calculation of the angle:
The change in momentum vector can be expressed as:
Δp = Δpxi + Δpyj
where Δpx and Δpy are the x-component and y-component of the change in momentum.
The angle between the x-axis and the change in momentum vector can be calculated using the arctangent function:
θ = arctan(Δpy / Δpx)
Now, let's plug in the given values and solve the problem step by step:
Given values:
m = 0.150 kg
v1 = 2.50 m/s
θ1 = 60.0°
v2 = 2.20 m/s
θ2 = 47.0°
a) Calculation of initial momentum (p1):
v1x = v1 * cos(θ1) = 2.50 * cos(60.0°) ≈ 2.50 * 0.5 ≈ 1.25 m/s
v1y = v1 * sin(θ1) = 2.50 * sin(60.0°) ≈ 2.50 * 0.866 ≈ 2.16 m/s
p1 = m * v1 = 0.150 * 2.50 ≈ 0.375 kg·m/s
b) Calculation of final momentum (p2):
v2x = v2 * cos(θ2) = 2.20 * cos(47.0°) ≈ 2.20 * 0.681 ≈ 1.50 m/s
v2y = v2 * sin(θ2) = 2.20 * sin(47.0°) ≈ 2.20 * 0.731 ≈ 1.61 m/s
p2 = m * v2 = 0.150 * 2.20 ≈ 0.330 kg·m/s
c) Calculation of the change in momentum (Δp):
Δp = p2 - p1 = 0.330 - 0.375 ≈ -0.045 kg·m/s
d) Calculation of the angle:
Δpx = p2x - p1x = 1.50 - 1.25 = 0.25 kg·m/s
Δpy = p2y - p1y = 1.61 - 2.16 ≈ -0.55 kg·m/s
θ = arctan(Δpy / Δpx) = arctan(-0.55 / 0.25) ≈ -67.58°
Therefore, the answers to the questions are:
a) The magnitude of the change in momentum of the billiard ball is approximately 0.045 kg·m/s.
b) The change in momentum vector points in the direction of approximately -67.58° with respect to the x-axis.