2-body collision: Min and Max speed

In summary, the conversation is about an elastic collision between two balls of different masses. The maximum speed of the second ball is found using the conservation of momentum and the center of mass frame. It is given by 2Mv/(M+m). The minimum speed is not specified, but it is mentioned that it will be lower than the maximum speed. The possibility of a non head-on collision is also discussed.
  • #1
unscientific
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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 VCM
step 2: u2' = u2 - VCM
step 2: pi = pf = 0
step 3: v2' = -u2
step 4: go back into lab frame by adding VCM

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..
 
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  • #2
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.
 
  • #3
mfb said:
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?
 
  • #5

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.
 

FAQ: 2-body collision: Min and Max speed

What is a 2-body collision?

A 2-body collision is a type of collision that involves two objects colliding with each other. In this scenario, the two objects are the only ones involved in the collision, making it a simpler and more manageable situation to analyze.

How is the minimum speed of a 2-body collision calculated?

The minimum speed of a 2-body collision is calculated using the conservation of momentum and energy equations. This involves determining the initial and final velocities of the objects, as well as their masses and the coefficient of restitution, which measures the elasticity of the collision.

What is the maximum speed in a 2-body collision?

The maximum speed in a 2-body collision is the speed at which the two objects collide with each other if they were to have a perfectly inelastic collision. This means that after the collision, the two objects would stick together and move with a common final velocity.

How does the mass of the objects affect the minimum and maximum speeds in a 2-body collision?

The mass of the objects has a direct impact on the minimum and maximum speeds in a 2-body collision. Generally, the larger the mass of the objects, the lower the minimum speed and the higher the maximum speed. This is because larger objects have more inertia and therefore require more energy to change their velocities.

Can the minimum and maximum speeds in a 2-body collision be equal?

No, the minimum and maximum speeds in a 2-body collision cannot be equal. This is due to the fact that the minimum speed is calculated based on the conservation of momentum and energy, while the maximum speed is determined by the coefficient of restitution. These two factors will always result in different values for the minimum and maximum speeds.

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