A 96.5 kg ice hockey player hits a 0.150 kg puck, giving the puck a velocity of 45.5 m/s. If both are initially at rest and if the ice is friction less, how far does the player recoil in the time it takes the puck to reach the goal 18.0 m away?
To find the distance the player recoils, we need to apply the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object is defined as the product of its mass and velocity. Mathematically, momentum (p) is given by the equation:
p = m * v
Where p is momentum, m is mass, and v is velocity.
Given information:
Mass of player (m1) = 96.5 kg
Mass of puck (m2) = 0.150 kg
Initial velocity of player (v1) = 0 m/s (player is initially at rest)
Initial velocity of puck (v2) = 0 m/s (puck is initially at rest)
Final velocity of puck (v2') = 45.5 m/s
Distance to the goal (d) = 18.0 m
Conservation of momentum equation:
(m1 * v1) + (m2 * v2) = (m1 * v1') + (m2 * v2')
Since the player is initially at rest (v1 = 0 m/s) and the puck is initially at rest (v2 = 0 m/s), the equation simplifies to:
(m1 * 0) + (m2 * 0) = (m1 * v1') + (m2 * v2')
Simplifying further, we have:
0 = (m1 * v1') + (m2 * v2')
Rearranging the equation to solve for the player's final velocity (v1'), we get:
v1' = -(m2 * v2') / m1
Now that we have the player's final velocity, we can determine the distance the player recoils using the equation of motion:
v1' = ∆x / ∆t
Where ∆x is the distance and ∆t is the time.
Since the player and puck are initially at rest, the recoil time ∆t is the same as the time taken by the puck to reach the goal.
We can calculate the time taken by the puck to reach the goal using the equation of motion:
d = v2' * ∆t
Rearranging the equation to solve for ∆t, we get:
∆t = d / v2'
Now we have the recoil time (∆t). Plugging this into the equation of motion for recoil, we can find the distance (∆x):
v1' = ∆x / ∆t
Rearranging the equation, we have:
∆x = v1' * ∆t
Substituting the known values, we can now calculate the distance the player recoils.