a pair of identical pucks move towards either other on a frictionless surface with equal speeds, the pucks move apart after colliding. during interaction 3/4 of energy is lost. what is the ratio (momentum of one puck before collision)/(momentum of one puck after the collision)?
To find the ratio of the momentum of one puck before the collision to the momentum of one puck after the collision, we need to analyze the conservation of momentum and the conservation of energy.
Let's assign variables for the various quantities involved:
m = mass of each puck
v = velocity of each puck before collision
v' = velocity of each puck after collision
C = coefficient of restitution (fraction of energy lost during the collision)
The conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision:
Total momentum before = Total momentum after
Now, let's calculate the initial momentum before the collision:
Momentum of one puck before = mass * velocity = m * v
Since there are two identical pucks with the same momentum, the total momentum before the collision is:
Total momentum before = 2 * (m * v)
Next, we need to consider the conservation of energy. According to the problem, 3/4 of the energy is lost during the collision. This means that the remaining energy is 1 - 3/4 = 1/4 of the initial energy.
Using the law of conservation of energy, we can connect it with the coefficient of restitution:
(Energy after collision) / (Energy before collision) = (Total momentum after collision)² / (Total momentum before collision)²
Replacing the terms with the given values:
(1/4) / 1 = [(2 * (m * v'))²] / [(2 * (m * v))²]
1 / 4 = (4 * (m * v')²) / (4 * (m * v)²)
1 / 4 = (m * v')² / (m * v)²
Taking the square root of both sides:
1 / 2 = (m * v') / (m * v)
Now, we can simplify the equation by canceling out the masses:
1 / 2 = v' / v
The ratio of the momentum of one puck before the collision to the momentum of one puck after the collision is 1:2.