2. In curling match, 6.0kg rock with speed 3.50 m/s collides with another motionless 6.0kg rock. What are the velocities of the rocks after the collision if it is (a) elastic (b) perfectly inelastic?
(a) Elastic collision:
Using the conservation of momentum in elastic collisions:
m1 * v1 initial + m2 * v2 initial = m1 * v1 final + m2 * v2 final
Where:
m1 = mass of first rock = 6.0 kg
v1 initial = initial velocity of first rock = 3.50 m/s
m2 = mass of second rock = 6.0 kg
v2 initial = initial velocity of second rock = 0 m/s (since it is motionless)
v1 final = final velocity of first rock (unknown)
v2 final = final velocity of second rock (unknown)
Plugging the values into the equation:
6.0 kg * 3.50 m/s + 6.0 kg * 0 m/s = 6.0 kg * v1 final + 6.0 kg * v2 final
21.0 kg m/s = 6.0 kg * v1 final + 6.0 kg * v2 final
Since kinetic energy is conserved in elastic collisions, we can also use the conservation of kinetic energy:
0.5 * m1 * (v1 initial)^2 + 0.5 * m2 * (v2 initial)^2 = 0.5 * m1 * (v1 final)^2 + 0.5 * m2 * (v2 final)^2
0.5 * 6.0 kg * (3.50 m/s)^2 = 0.5 * 6.0 kg * (v1 final)^2 + 0
36.75 kg m^2/s^2 = 3.0 kg * (v1 final)^2
12.25 m^2/s^2 = (v1 final)^2
v1 final ≈ √12.25 m/s = 3.50 m/s
Then we can use this result to find v2 final:
21.0 kg m/s = 6.0 kg * 3.50 m/s + 6.0 kg * v2 final
21.0 kg m/s - 21.0 kg m/s = 6.0 kg * v2 final
0 kg m/s = 6.0 kg * v2 final
v2 final = 0 m/s
Therefore, in an elastic collision, the final velocities of the rocks are:
v1 final = 3.50 m/s
v2 final = 0 m/s
(b) Perfectly inelastic collision:
In a perfectly inelastic collision, the two rocks stick together after colliding, so their final velocities will be the same.
Using the conservation of momentum for perfectly inelastic collisions::
m1 * v1 initial + m2 * v2 initial = (m1 + m2) * v final
Plugging the values into the equation:
6.0 kg * 3.50 m/s + 6.0 kg * 0 m/s = (6.0 kg + 6.0 kg) * v final
21.0 kg m/s = 12.0 kg * v final
v final = 1.75 m/s
Therefore, in a perfectly inelastic collision, the final velocity of the rocks is:
v final = 1.75 m/s