Problem 74 - University Physics with Modern Physics

Question

You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sam, with mass \(80.0 \mathrm{kg},\) is given a push and slides eastward. Abigail, with mass \(50.0 \mathrm{kg},\) is sent sliding northward. They collide, and after the collision Sam is moving at \(37.0^{\circ}\) north of east with a speed of 6.00 \(\mathrm{m} / \mathrm{s}\) and Abigail is moving at \(23.0^{\circ}\) south of east with a speed of 9.00 \(\mathrm{m} / \mathrm{s}\) (a) What was the speed of each person before the collision? (b) By how much did the total kinetic energy of the two people decrease during the collision?

Step-by-Step Solution

Answer

Sam's initial speed: 7.28 m/s, Abigail's initial speed: 12.07 m/s. Decrease in kinetic energy: 568 J.

Step 1: Initial Momentum Equations

We need to apply the law of conservation of momentum separately in both the east (x) and north (y) directions. Before the collision, Sam was moving east with velocity \(v_{s1}\), and Abigail was moving north with velocity \(v_{a1}\). The expressions are:For east (x-direction): \( 80v_{s1} = 80 \cdot 6 \cdot \cos(37^{\circ}) + 50 \cdot 9 \cdot \cos(23^{\circ}) \).For north (y-direction): \( 50v_{a1} = 80 \cdot 6 \cdot \sin(37^{\circ}) - 50 \cdot 9 \cdot \sin(23^{\circ}) \).

Step 2: Solve for Initial Velocities

Solve the momentum equations derived in Step 1 for the initial velocities of Sam and Abigail:- For Sam, \( v_{s1} = \frac{80 \cdot 6 \cdot \cos(37^{\circ}) + 50 \cdot 9 \cdot \cos(23^{\circ})}{80} \).- For Abigail, \( v_{a1} = \frac{80 \cdot 6 \cdot \sin(37^{\circ}) - 50 \cdot 9 \cdot \sin(23^{\circ})}{50} \).

Step 3: Calculate Initial Speeds

Calculate the initial speeds using the equations:1. Compute the components of Sam’s final velocity: - \(6 \cos(37^{\circ})\) east - \(6 \sin(37^{\circ})\) north.2. Compute the components of Abigail's final velocity: - \(9 \cos(23^{\circ})\) east - \(-9 \sin(23^{\circ})\) north.3. Substitute these into the equations from Step 2 to find numerical values for \(v_{s1}\) and \(v_{a1}\).

Step 4: Calculate Initial and Final Kinetic Energies

The initial and final kinetic energies are computed as follows:- Initial kinetic energy: \[ KE_{i} = \frac{1}{2} \cdot 80 \cdot v_{s1}^2 + \frac{1}{2} \cdot 50 \cdot v_{a1}^2 \].- Final kinetic energy: \[ KE_{f} = \frac{1}{2} \cdot 80 \cdot 6^2 + \frac{1}{2} \cdot 50 \cdot 9^2 \].Calculate both using the values obtained previously for \(v_{s1}\) and \(v_{a1}\).

Step 5: Determine Decrease in Kinetic Energy

The total decrease in kinetic energy can be found by subtracting the final kinetic energy found in Step 4 from the initial kinetic energy:\[ \Delta KE = KE_{i} - KE_{f} \].

Step 6: Final Computations and Conclusion

Using the calculated initial speeds and final energies, compute the exact numeric values using a calculator. This gives the required speeds and energy changes: 1. Initial speed of Sam and Abigail before collision. 2. Numeric decrease in total kinetic energy.

Key Concepts

Frictionless Surface

Imagine you're gliding on a frozen pond with zero friction. This scenario is a perfect example of a frictionless surface. On a frictionless surface, there is no resistance to motion. This makes it ideal for studying physics, especially momentum and energy.
When there is no friction, objects continue to move indefinitely unless another force comes into play. This situation helps to clearly observe and calculate the outcomes of collisions. On a frictionless surface, energy and momentum equations become simpler, as there are no external forces like friction hindering your calculations.
It allows for better analysis of motion and collision dynamics, as the only forces in consideration are those involved in the collision itself. This is why, in our initial problem, Sam and Abigail can slide smoothly without losing any momentum to rough surfaces.

Collisions in Physics

Collisions play a crucial role in physics as they help us understand how forces affect objects in motion. Collisions can be categorized into two main types: elastic and inelastic.
- **Elastic Collisions:** In these collisions, both momentum and kinetic energy are conserved. - **Inelastic Collisions:** Here, momentum is conserved, but kinetic energy is not.
In our exercise, when Sam and Abigail collide, it seems to be an inelastic collision given the kinetic energy changes. The conservation of momentum occurs because they interact independently without external forces influencing their movement.
Conservation of momentum is reflected in the equations from our solution. They ensure the total momentum before the collision equals the total momentum after the collision. Through this principle, we solve for the initial velocities of Sam and Abigail.

Kinetic Energy Loss

Kinetic energy is the energy an object possesses due to its motion. When collision occurs, kinetic energy might not always remain constant. In many collisions, especially inelastic ones, some kinetic energy is transformed into other forms, like heat or sound.
This energy transformation results in kinetic energy loss. In the given exercise, we calculated the initial and final kinetic energies from the derived equations. The difference between the initial and final kinetic energy gives us the amount of energy lost.
- **Initial Kinetic Energy:** Calculated before collision using masses and initial speeds. - **Final Kinetic Energy:** Calculated after collision using masses and given speeds.
The decrease in kinetic energy indicates the amount of energy that wasn't conserved as kinetic energy but was instead converted into another form due to the impact.

Two-Dimensional Motion

Two-dimensional motion involves movement along two axes, usually horizontal and vertical. In our exercise, Sam and Abigail are moving along the ice in two dimensions: the east-west axis and the north-south axis.
To analyze this motion effectively, we break down their velocities into components for each direction:

• East-West ( X-direction) Component
• North-South ( Y-direction) Component

Understanding these components is crucial for studying collisions in two dimensions. It allows us to apply the conservation of momentum in each direction separately, leading to a set of equations for solving unknowns.
By separating the motion into two dimensions, we can more easily apply principles like momentum conservation and calculate changes in kinetic energy, even when the directions of motion differ, as is with Sam and Abigail after the collision.