Question

A 70.0-kg ice hockey goalie, originally at rest, catches a 0.150-kg hockey puck slapped at him at a velocity of 35.0 m/s. Suppose the goalie and the ice puck have an elastic collision and the puck is reflected back in the direction from which it came. What would their final velocities be in this case?

v_puck =

v_goalie =

Answer 1

Let the mass of the hockey goalie be $m_1$ and the mass of the hockey puck be $m_2$. Let the initial speed of the hockey goalie be $u_1$ and the initial speed of the hockey puck be $u_2$. From the law of conservation of momentum, we have:

$$m_1u_1+m_2u_2=m_1v_1+m_2v_2. (1)$$

Since collision is elastic, kinetic energy is conserved and we can write:

$$\dfrac{1}{2}m_1u_1^2+\dfrac{1}{2}m_2u_2^2=\dfrac{1}{2}m_1v_1^2+\dfrac{1}{2}m_2v_2^2. (2)$$

Let’s rearrange equations (1) and (2):

$$m_1(u_1-v_1)=m_2(v_2-u_2), (3)$$ $$m_1(u_1^2-v_1^2)=m_2(v_2^2-u_2^2). (4)$$

Let’s divide equation (4) by equation (3):

$$\dfrac{(u_1-v_1)(u_1+v_1)}{u_1-v_1}=\dfrac{(v_2-u_2)(v_2+u_2)}{v_2-u_2},$$ $$u_1+v_1=u_2+v_2. (5)$$

Let's express $v_2$ from the equation (5) in terms of $u_1$, $u_2$ and $v_1$:

$$v_2=u_1-u_2+v_1. (6)$$

Let’s substitute equation (6) into equation (3). After simplification, we get:

$$(m_1-m_2)u_1+2m_2u_2=(m_1+m_2)v_1.$$

From this equation we can find the final speed of the hockey goalie, $v_1$:

$$v_1=\dfrac{(m_1-m_2)}{(m_1+m_2)}u_1+\dfrac{2m_2}{(m_1+m_2)}u_2.$$

Since, $u_1=0$ (hockey goalie, originally at rest), we get:

$$v_1=\dfrac{(m_1-m_2)}{(m_1+m_2)}u_1+\dfrac{2m_2}{(m_1+m_2)}u_2.$$ $$v_1=\dfrac{2\cdot0.150\ kg}{(70\ kg+0.150\ kg)}\cdot(-35.0\ \dfrac{m}{s})=-0.150\ \dfrac{m}{s}.$$

The sign minus means that the hockey goalie moves to the left after collision.

Substituting $v_1$into the equation (6) we can find the final speed of the hockey puck, $v_2$:

$$v_2=\dfrac{2m_1}{(m_1+m_2)}u_1+\dfrac{(m_2-m_1)}{(m_1+m_2)}u_2,$$ $$v_2=\dfrac{(m_2-m_1)}{(m_1+m_2)}u_2,$$ $$v_2=\dfrac{(0.150\ kg-70\ kg)}{(0.150\ kg+70\ kg)}\cdot(-35\ \dfrac{m}{s})=34.9\ \dfrac{m}{s}.$$

The sign plus means that the hockey puck moves to the right after collision.