Problem 15 - College Physics

Question

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

Step-by-Step Solution

Answer

Daniel's velocity is 5.53 m/s at 46.9° to x-axis. Change in kinetic energy is -626 J.

Step 1: Understand Momentum Conservation

In a collision where no external forces act, the total momentum before the collision equals the total momentum after the collision. For this scenario, both skaters' movements define a two-dimensional plane. Use the conservation of momentum principle in both the x and y components.

Step 2: Calculate Initial and Final Momentum in the X-direction

The initial momentum of Rebecca in the x-direction is \( p_{x_i} = m_R \cdot v_{R_i} \) where \( m_R = 45.0 \, \mathrm{kg} \) and \( v_{R_i} = 13.0 \, \mathrm{m/s} \). Since Daniel is initially at rest, his initial momentum is zero. Hence, the total initial momentum in the x-direction is \( p_{x_{initial}} = 45.0 \, \mathrm{kg} \times 13.0 \, \mathrm{m/s} = 585.0 \, \mathrm{kg\cdot m/s} \). After the collision, Rebecca's x-component of velocity is \( v_{Rx_f} = 8.00 \, \mathrm{m/s} \times \cos(53.1^\circ) \). Her final x-momentum is \( p_{xR_final} = 45.0 \, \mathrm{kg} \times v_{Rx_f} \). Daniel's final x-momentum is \( 65.0 \, \mathrm{kg} \times v_{Dx} \). Set the initial total x-momentum equal to the final total x-momentum and solve for \( v_{Dx} \).

Step 3: Calculate Momentum in the Y-direction

The initial momentum in the y-direction is zero, as Rebecca moves solely in the x-direction initially. After collision, Rebecca's y-component of velocity is \( v_{Ry_f} = 8.00 \, \mathrm{m/s} \times \sin(53.1^\circ) \). Her final y-momentum is \( p_{yR_final} = 45.0 \, \mathrm{kg} \times v_{Ry_f} \). Similarly, Daniel's y-momentum is \( 65.0 \, \mathrm{kg} \times v_{Dy} \). Setting these equal gives us a relation to solve for \( v_{Dy} \).

Step 4: Solve for Daniel's Final Velocity

Using the results from the momentum calculations in Steps 2 and 3, determine Daniel's velocity components \( v_{Dx} \) and \( v_{Dy} \). Then find the magnitude of his velocity \( v_D \) using \( v_D = \sqrt{v_{Dx}^2 + v_{Dy}^2} \) and the direction \( \theta = \tan^{-1}\left(\frac{v_{Dy}}{v_{Dx}}\right) \).

Step 5: Kinetic Energy Change Calculation

Compute the initial kinetic energy of Rebecca using \( KE_i = \frac{1}{2} m_R v_{R_i}^2 \). After collision, compute the kinetic energy of both Rebecca and Daniel as \( KE_{R_f} = \frac{1}{2} m_R v_{R_f}^2 \) and \( KE_{D_f} = \frac{1}{2} m_D v_D^2 \) respectively. The change in total kinetic energy is \( \Delta KE = (KE_{R_f} + KE_{D_f}) - KE_i \).

Key Concepts

Collisions

Collisions are events where two or more bodies exert forces on each other for a relatively short time. They can be categorized primarily into elastic and inelastic collisions. In an elastic collision, both momentum and kinetic energy are conserved, while in an inelastic collision, only momentum is conserved.
In the scenario with our ice skaters, the collision is inelastic. This is because, even though momentum is conserved (total momentum before equals total momentum after), the kinetic energy is not.

• This means that some of the kinetic energy is transformed into other forms of energy, such as heat or sound, usually due to friction or deformation during the impact.
• Understanding these basics of collisions helps explain why Daniel and Rebecca individually have different velocities post-collision.

Kinetic Energy

Kinetic energy is the energy of motion. It is given by the formula:\[ KE = \frac{1}{2} m v^2 \] where

• \(m\) is the mass of the object, and
• \(v\) is its velocity.

In the context of the exercise, we're interested in how kinetic energy changes as a result of the collision. Initially, only Rebecca has kinetic energy because she is moving at 13.0 m/s, whereas Daniel is at rest. This initial kinetic energy is given by: \[ KE_i = \frac{1}{2} \times 45.0 \, \text{kg} \times (13.0 \, \text{m/s})^2 \]After the collision, both skaters have some kinetic energy, but as the initial kinetic energy is partly converted into other forms, the total kinetic energy decreases. Hence,

• the change in kinetic energy (ΔKE) results from calculating the initial kinetic energy and subtracting the combined final kinetic energies of both skaters.

Ice Skaters

In physics problems involving ice skaters, the lack of friction is often highlighted. This makes ice skating examples ideal for studying principles like momentum conservation and kinetic energy within a closed, isolated system.
Rebecca and Daniel are on ice, meaning there are minimal external forces affecting their motion.
This allows us to apply conservation laws directly and observe pure two-body collisions.

• The mass of Rebecca and Daniel plays a crucial role in determining their subsequent velocities post-collision.
• By understanding the masses and initial velocities, we can solve for unknown parameters like the final velocities in each component direction.

Two-Dimensional Motion

Two-dimensional motion occurs when an object moves in a plane, requiring both x and y components for full description. This is opposed to linear motion, which involves only one dimension.
In the skaters' problem, Rebecca initially moves in the x-direction, and during the collision, her velocity changes magnitude and direction.

• To find the post-collision velocities, we must consider both directions. This is achieved using components like \(v_{x}\) and \(v_{y}\), computed using trigonometric functions based on the angle.
• This leads to the use of vector addition and trigonometry to fully describe the motion.

By solving the momentum equations for both dimensions separately, we can determine each skater's final motion parameters.