A 98.0 kg ice hockey player hits a 0.150 kg puck, giving the puck a velocity of 49.5 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 13.5 m away?
To determine how far the player recoils, we can apply the principle of conservation of momentum.
The equation for momentum is given by:
p = m * v
where p is the momentum, m is the mass, and v is the velocity.
Initially, both the player and the puck are at rest, so the total momentum before the collision is zero. After the collision, the puck has a velocity of 49.5 m/s, so its momentum is:
p_puck = m_puck * v_puck
Substituting the given values, we have:
p_puck = (0.150 kg) * (49.5 m/s)
= 7.425 kg·m/s
According to the conservation of momentum, the total momentum after the collision should also be zero. This means the player must have a velocity in the opposite direction of the puck.
Let's denote the velocity of the player as v_player, then:
p_player = m_player * (-v_player)
The negative sign is used because the velocities of the player and the puck are in opposite directions.
The total momentum after the collision is:
p_total = p_puck + p_player
= 7.425 kg·m/s + (-m_player * v_player)
= 0
From this equation, we can solve for v_player:
-v_player = -7.425 kg·m/s / m_player
Now, we know the formula for displacement is given by:
d = v * t
where d is the displacement, v is the velocity, and t is the time.
In this case, we need to determine the time it takes for the puck to reach the goal, given a displacement of 13.5 m and a velocity of 49.5 m/s.
Using the formula:
t = d / v_puck
t = 13.5 m / 49.5 m/s
t = 0.2727 s (rounded to four decimal places)
Now, we have the time, t, it took for the puck to reach the goal.
To find the player's displacement, we can use the equation for displacement:
d_player = v_player * t
Plugging in the known values:
d_player = -v_player * t
= -((-7.425 kg·m/s) / m_player) * 0.2727 s
This gives us the player's recoil distance in terms of the player's mass, m_player. Unfortunately, we don't have the mass of the player, so we can't determine the exact recoil distance without that information.