Question

Two ice skaters, Daniel (mass 65.0 kg) and Rebecca (mass 45.0 kg), are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 m/s before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 m/s at an angle of 53.1° from her initial direction. Both skaters move on the frictionless, horizontal surface of the rink. What is the change in total kinetic energy of the two skaters as a result of the collision?

Accepted Answer

Step 1: Identify the type of collision. Since the problem involves two skaters colliding and moving in different directions afterward, this is an example of a two-dimensional inelastic collision. Momentum is conserved in both the x and y directions, but kinetic energy may not be conserved. Step 2: Write the conservation of momentum equations. Momentum is conserved in both the x and y directions. Let Daniel's velocity after the collision be $$v_{D,x}$$ and $$v_{D,y} in $$the x and y directions, respectively. Rebecca's velocity components after the collision are $$v_{R,x} = 8.00 \cos(53.1°)$$ and $$v_{R,y} = 8.00 \sin(53.1°). $$The equations for momentum conservation are: $$m_R v_{R,i} = m_R v_{R,x} + m_D v_{D,x} (x-$$direction) and $$0 = m_R v_{R,y} + m_D v_{D,y} (y-$$direction), where $$m_R$$ and $$m_D$$ are the masses of Rebecca and Daniel, respectively, and $$v_{R,i} is $$Rebecca's initial velocity. Step 3: Solve for Daniel's velocity components. Rearrange the x-direction momentum equation to find $$v_{D,x}$$: $$v_{D,x} = \frac{m_R (v_{R,i} - v_{R,x})}{m_D}. $$Similarly, rearrange the y-direction momentum equation to find $$v_{D,y}$$: $$v_{D,y} = -\frac{m_R v_{R,y}}{m_D}. $$Substitute the known values for $$m_R$$, $$m_D$$, $$v_{R,i}$$, and the components of $$v_R to $$calculate $$v_{D,x}$$ and $$v_{D,y}. $$Step 4: Determine the magnitude and direction of Daniel's velocity. The magnitude of Daniel's velocity is given by $$v_D = \sqrt{v_{D,x}^2 + v_{D,y}^2}. $$The direction (angle $$	heta) of $$his velocity relative to the x-axis is $$	heta = 	an^{-1}\left(\frac{v_{D,y}}{v_{D,x}}ight). $$Step 5: Calculate the change in total kinetic energy. The initial kinetic energy is $$KE_{initial} = \frac{1}{2} m_R v_{R,i}^2$$, as Daniel is initially at rest. The final kinetic energy is $$KE_{final} = \frac{1}{2} m_R (v_{R,x}^2 + v_{R,y}^2) + \frac{1}{2} m_D (v_{D,x}^2 + v_{D,y}^2). $$The change in kinetic energy is $$\Delta KE = KE_{final} - KE_{initial}. $$Substitute the known values to compute $$\Delta KE.$$

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Key Concepts

Conservation of Momentum

Kinetic Energy

Vector Components