2-body collision: Min and Max speed

[unscientific]

Homework Statement

A ball of mass M traveling at non-relativistic speed v elastically collides with a stationary ball of mass m.

(a)Show that the maximum speed which the second ball can have is:

2Mv/(M+m)(b)What is the minimum speed?

The Attempt at a Solution

(a)
Method 1

Go into ZMF frame:

step 1: find V_CM
step 2: u₂' = u₂ - V_CM
step 2: p_i = p_f = 0
step 3: v₂' = -u₂
step 4: go back into lab frame by adding V_CM

Method 2

1)Speed of approach = speed of separation
2)conservation of momentum
3)done.

But the puzzling thing is that why is this the maximum speed? Isn't this the actual acquired speed?

(b)
Do we assume a head-on collision? If it is a head-on collision won't the speed be the same as part (a)?

assuming a non head-on collision by including angles and stuff doesn't help..

 

[mfb]

If you assume a head-on collision where the second ball travels in the initial direction of ball 1, this is the maximal speed.
Every other collision will give a lower speed.

 

[unscientific]

If you assume a head-on collision where the second ball travels in the initial direction of ball 1, this is the maximal speed.
Every other collision will give a lower speed.

so for part (b) it isn't a head-on collision?

 

[mfb]

Right ;)

 

[paranormal]

I can provide an explanation for the maximum and minimum speeds in a 2-body collision. In an elastic collision, kinetic energy is conserved, meaning the total kinetic energy before and after the collision remains the same. In this scenario, the maximum speed of the second ball (m) can acquire is when it moves in the same direction as the first ball (M) with the same velocity (v). This is because the first ball is transferring all its kinetic energy to the second ball, causing it to reach its maximum speed. This can be calculated using the equation 2Mv/(M+m), as shown in part (a) of the problem.

On the other hand, the minimum speed of the second ball can acquire is when it moves in the opposite direction of the first ball with the same velocity (v). In this scenario, the first ball is transferring all its kinetic energy to the second ball, causing it to reach its minimum speed. This minimum speed can be calculated using the same equation, but with the negative sign, as the second ball is now moving in the opposite direction. This is known as the head-on collision scenario.

However, if the collision is not head-on and there are angles involved, the minimum speed of the second ball can vary depending on the direction and magnitude of the angles. In this case, the minimum speed cannot be calculated using the same equation as in the head-on collision scenario. Instead, the conservation of momentum and kinetic energy equations must be used to determine the minimum speed.

In conclusion, the maximum speed in a 2-body collision is the actual acquired speed of the second ball when it moves in the same direction as the first ball, while the minimum speed is the actual acquired speed when the second ball moves in the opposite direction of the first ball. The specific values for these speeds can vary depending on the scenario, but the equations and principles of conservation of momentum and kinetic energy remain the same.