Initially, mass one (2.10 kg) has a velocity of 5.90 m/s and mass two (2.80 kg) is at rest. After they collide, mass one emerges at an angle theta = 33.0 degrees. What is the speed of mass one after the collision if the collision is completely elastic? (Note, there are actually two possible answers two this problem, choose the solution which has m1 going as fast as possible.)
What is the angle phi, between mass two's velocity and the initial velocity of mass one? (Give your answer as a positive number in degrees.)
What is the final speed of mass two after the collision?
To solve these questions, we can use the principle of conservation of momentum and conservation of kinetic energy.
1. Let's start by finding the speed of mass one after the collision. The conservation of momentum equation can be written as:
Initial momentum in x-direction = Final momentum in x-direction
m1 * v1x_initial = m1 * v1x_final + m2 * v2x_final
Since mass two is initially at rest, we can rewrite the equation as:
m1 * v1x_initial = m1 * v1x_final
Plugging in the values:
(2.10 kg) * (5.90 m/s) = (2.10 kg) * v1x_final
Solving for v1x_final:
v1x_final = (2.10 kg * 5.90 m/s) / (2.10 kg) = 5.90 m/s
So, the speed of mass one after the collision is 5.90 m/s.
2. To find the angle phi between mass two's velocity and the initial velocity of mass one, we need to find the components of both velocities in the x and y directions.
The initial velocity of mass one can be written as:
v1_initial = v1x_initial * cos(theta) (in x-direction) + v1y_initial * sin(theta) (in y-direction)
Since mass two is at rest, its velocity can be written as:
v2_initial = 0 (in x-direction) + 0 (in y-direction)
The angle phi can be found using the dot product of the velocities:
cos(phi) = (v1x_initial * v2x_initial + v1y_initial * v2y_initial) / (|v1_initial| * |v2_initial|)
We know that v2_initial = 0, so the equation becomes:
cos(phi) = v1x_initial / |v1_initial|
Plugging in the values:
cos(phi) = 5.90 m/s / sqrt((5.90 m/s)^2 + 0^2)
Solving for phi:
phi = arccos(5.90 m/s / sqrt((5.90 m/s)^2)) = 0 degrees
Therefore, the angle phi between mass two's velocity and the initial velocity of mass one is 0 degrees.
3. To find the final speed of mass two after the collision, we can use the conservation of kinetic energy equation:
Initial kinetic energy = Final kinetic energy
(1/2) * m2 * v2_initial^2 = (1/2) * m2 * v2_final^2
Since mass two is initially at rest, the equation simplifies to:
0 = (1/2) * m2 * v2_final^2
So, the final speed of mass two after the collision is 0 m/s.
To summarize:
- The speed of mass one after the collision is 5.90 m/s.
- The angle phi between mass two's velocity and the initial velocity of mass one is 0 degrees.
- The final speed of mass two after the collision is 0 m/s.