A white billiard ball with mass mw = 1.43 kg is moving directly to the right with a speed of v = 3.4 m/s and collides elastically with a black billiard ball with the same mass mb = 1.43 kg that is initially at rest. The two collide elastically and the white ball ends up moving at an angle above the horizontal of θw = 39° and the black ball ends up moving at an angle below the horizontal of θb = 51°.
a. What is the final speed of the white ball?
b. What is the final speed of the black ball?
c. What is the magnitude of the final total momentum of the system?
d. What is the final total energy of the system?
a. Vw = 3.4 cos(39)
= 2.64 m/s
b. Vb = 3.4 cos(51)
= 2.14 m/s
c. mVo = 1.43 kg * 3.4 m/s
= 4.86 kg*m/s
d.E = 1/2 m v^2 = 1/2 (1.43) (3.4^2)
=8.27 J
a. The final speed of the white ball can be calculated using the formula for momentum conservation in an elastic collision:
mw * v = mw * vw + mb * vb
where vw is the final speed of the white ball and vb is the final speed of the black ball. Rearranging the equation, we can solve for vw:
vw = (mw * v - mb * vb) / mw
b. Similarly, the final speed of the black ball can be calculated using the same equation:
vb = (mw * v - mb * vw) / mb
c. The magnitude of the final total momentum of the system can be calculated by summing the individual momenta of the white and black balls:
total momentum = mw * vw + mb * vb
d. The final total energy of the system can be calculated using the formula for kinetic energy:
total energy = 0.5 * mw * vw^2 + 0.5 * mb * vb^2
To solve this problem, we can use the conservation of momentum and the conservation of energy.
a. To find the final speed of the white ball, we can use the conservation of momentum. The initial momentum of the system is zero since the black ball is initially at rest.
Let's define the positive x-direction as the direction of the initial motion of the white ball. The initial momentum of the white ball is:
Pwi = mw * v
The final momentum of the white ball can be broken down into horizontal and vertical components:
Pwf_x = mw * vf
Pwf_y = mw * vf * sin(θw)
The momentum conservation equation in the x-direction gives us:
Pwi = Pwf_x
mw * v = mw * vf
Solving for vf:
vf = v
So, the final speed of the white ball is 3.4 m/s.
b. To find the final speed of the black ball, we can use the conservation of momentum. The initial momentum of the black ball is zero since it is initially at rest.
Let's define the positive y-direction as the direction opposite to the initial motion of the white ball. The initial momentum of the black ball is:
Pbi = 0
The final momentum of the black ball can be broken down into horizontal and vertical components:
Pbf_x = mb * vf * cos(θb)
Pbf_y = -mb * vf * sin(θb)
The momentum conservation equation in the y-direction gives us:
Pbi = Pbf_y
0 = -mb * vf * sin(θb)
Solving for vf:
vf = 0
So, the final speed of the black ball is 0 m/s.
c. The total momentum of the system is conserved. Since the initial momentum of the system is zero, the final total momentum is also zero.
d. The total mechanical energy of the system is conserved since the collision is elastic. The initial kinetic energy of the system is:
KEi = (1/2) * mw * v^2
The final kinetic energy of the system can be calculated by summing the kinetic energies of the white and black balls:
KEw = (1/2) * mw * vf^2
KEb = (1/2) * mb * vf^2
The final total kinetic energy is:
KEf = KEw + KEb
Since the collision is elastic, the final total kinetic energy is equal to the initial kinetic energy:
KEf = KEi
(1/2) * mw * vf^2 + (1/2) * mb * vf^2 = (1/2) * mw * v^2
Substituting the values:
(1/2) * 1.43 kg * vf^2 + (1/2) * 1.43 kg * vf^2 = (1/2) * 1.43 kg * (3.4 m/s)^2
Simplifying the equation:
1.43 kg * vf^2 + 1.43 kg * vf^2 = 1.43 kg * (3.4 m/s)^2
2.86 kg * vf^2 = 1.43 kg * (3.4 m/s)^2
vf^2 = (1.43 kg * (3.4 m/s)^2) / 2.86 kg
vf^2 = 5.2394 m^2/s^2
vf ≈ 2.29 m/s
So, the final kinetic energy and total energy of the system is the same as the initial kinetic energy, and the final total energy is given by:
KEf = (1/2) * mw * vf^2 + (1/2) * mb * vf^2
= (1/2) * 1.43 kg * (2.29 m/s)^2 + (1/2) * 1.43 kg * (2.29 m/s)^2
= 4.97 J (rounded to two decimal places)
Therefore, the final total energy of the system is approximately 4.97 J.
To solve this problem, we can use the principles of conservation of momentum and conservation of kinetic energy.
a. To find the final speed of the white ball, we need to break down the initial and final velocities into their horizontal and vertical components. We can use the following equations:
Initial momentum (before collision):
P_initial = mw * v
Final momentum (after collision):
P_final = mw * vw
To find vw, we can use the following equations:
Horizontal component of initial velocity:
vi_x = v * cos(0)
where cos(0) = 1 (since the ball is moving directly to the right)
Vertical component of initial velocity:
vi_y = v * sin(0) = 0 (since the initial vertical component is 0, the ball is not moving up or down)
Horizontal component of final velocity:
vf_x = vw * cos(θw)
Vertical component of final velocity:
vf_y = vw * sin(θw)
To find vw, we can use the Pythagorean theorem:
vw = √(vf_x² + vf_y²)
Substituting the given values:
vw = √((vf * cos(θw))² + (vf * sin(θw))²)
b. Similar to part a, we can apply the same equations to find the final speed of the black ball. Since the black ball is initially at rest, its velocity components will be:
Horizontal component of final velocity:
vf_x = vb * cos(θb)
Vertical component of final velocity:
vf_y = vb * sin(θb)
Using the Pythagorean theorem, we can find vb:
vb = √(vf_x² + vf_y²)
c. The total momentum of the system before the collision is:
P_initial_total = P_initial_white + P_initial_black
P_initial_total = mw * v
The total momentum of the system after the collision is:
P_final_total = P_final_white + P_final_black
P_final_total = mw * vw + mb * vb
d. The total initial kinetic energy of the system is given by:
KE_initial_total = KE_initial_white + KE_initial_black
KE_initial_total = 0.5 * mw * v²
The total final kinetic energy of the system is given by:
KE_final_total = KE_final_white + KE_final_black
KE_final_total = 0.5 * mw * vw² + 0.5 * mb * vb²
Now that we have the necessary equations, we can plug in the values to find the answers.