Solve Elastic Collisions Homework: Ball 2 Skid Distance

[Mastablade]

Homework Statement

Consider a pool table that contains two identical balls. Ball 1 is rolling at a speed of 1.1 m/sec when it hits ball 2 which is at rest. The coefficients of friction between the balls and the table are 0.8 (potential) and .5 (kinetic). The collision is perfectly elastic and there is no friction between the balls. How far will ball 2 skid before it is rolling without sliding.

Homework Equations

Speed - (Friction * Kinetic Energy)

The Attempt at a Solution

This seems rather simple, but I would like to check my work. Simply I need to find when the ball has overcome the force of friction, so wouldn't it be,
1.1 (velocity) - 9.8(friction) * .5 (kinetic energy) = 0?

if so that would turn out to be

t = 1.1/4.9 = .22 seconds

putting that back into the equation

1.1 = 4.9 * (.22) = .022 m

I am not quite sure if this is the correct approach to this problem, so help would be appreciated. Thanks in advance.

 

[anathema]

Thank you for your question. Your approach to this problem is correct, but I would like to provide a more detailed explanation.

First, let's define some variables:

- v1 = initial velocity of ball 1 (1.1 m/s)
- v2 = initial velocity of ball 2 (0 m/s)
- u1 = final velocity of ball 1 after the collision
- u2 = final velocity of ball 2 after the collision
- m = mass of each ball (we can assume they are the same)
- g = acceleration due to gravity (9.8 m/s^2)
- μp = coefficient of potential friction (0.8)
- μk = coefficient of kinetic friction (0.5)

Now, let's break down the problem into two parts: the collision and the skidding.

1. Collision:
Since the collision is perfectly elastic, we can use the conservation of momentum and conservation of kinetic energy equations:

Conservation of momentum: m*v1 + m*v2 = m*u1 + m*u2
Conservation of kinetic energy: 0.5*m*v1^2 + 0.5*m*v2^2 = 0.5*m*u1^2 + 0.5*m*u2^2

Substituting in our known values, we get:

m*1.1 + m*0 = m*u1 + m*u2
0.5*m*(1.1)^2 + 0.5*m*(0)^2 = 0.5*m*u1^2 + 0.5*m*u2^2

Solving for u1 and u2, we get:

u1 = 0.55 m/s
u2 = 0.55 m/s

This means that both balls will have the same velocity after the collision.

2. Skidding:
Now, we can use the equation for frictional force to determine how long it will take for ball 2 to overcome the force of friction and start rolling without sliding:

Ffr = μk * m * g

Since we know that the frictional force is equal to the change in kinetic energy, we can set the two equations equal to each other:

Ffr = μk * m * g = 0.5*m*u2^2

Solving for u2, we get:

u2 = √(2*μk*g