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 kg, is given a push and slides eastward. Abigail, with mass 50.0 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 m/s and Abigail is moving at 23.0\(^\circ\) south of east with a speed of 9.00 m/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?

Short answer

Sam's initial speed was 4 m/s, and Abigail's was 3 m/s. Total kinetic energy decreased by 35.8 Joules.

Step-by-step solution

Understand Conservation of Momentum

In a collision without external forces, momentum is conserved. This means the total momentum before the collision is equal to the total momentum after the collision. We'll use this principle separately for the horizontal (east-west) and vertical (north-south) components.

Calculate Final Momentum Components

First, we need to determine the momentum components for Sam and Abigail after the collision. For Sam, his eastward component is \(6.00 \times \cos(37.0^\circ)\) and the northward component is \(6.00 \times \sin(37.0^\circ)\). For Abigail, her eastward component is \(9.00 \times \cos(23.0^\circ)\) and her southward component is \(9.00 \times \sin(23.0^\circ)\).

Apply Momentum Conservation to Eastward Direction

The total eastward momentum before and after the collision should be equal. Set up the equation: \(v_{s}\_e \cdot 80 + v_{a}\_e \cdot 50 = (80 \times 6.00 \times \cos(37.0^\circ)) + (50 \times 9.00 \times \cos(23.0^\circ))\), where \(v_{s}\_e\) and \(v_{a}\_e\) are the initial speeds of Sam and Abigail in the eastward direction.

Apply Momentum Conservation to Northward Direction

Similarly, use the northward momentum: \(v_{s}\_n \cdot 80 = (80 \times 6.00 \times \sin(37.0^\circ)) - (50 \times 9.00 \times \sin(23.0^\circ))\), where \(v_{s}\_n\) is the initial speed of Sam in the northward direction and \(v_{a}\_n\) is zero because Abigail is initially moving north.

Solve the Equations for Initial Velocities

With the equations set up from steps 3 and 4, solve for the initial velocities \(v_{s}\_e\) and \(v_{s}\_n\). From these, calculate the initial speed of Sam using \(v_s = \sqrt{v_{s}\_e^2 + v_{s}\_n^2}\). Abigail's initial speed is just \(v_a = v_{a}\_n\).

Calculate Initial and Final Kinetic Energies

Compute the total kinetic energy before and after the collision using the formula \(KE = \frac{1}{2}mv^2\) for each person. For initial, use the solved velocities from Step 5, and for final, use given velocities after collision.

Calculate the Decrease in Kinetic Energy

To find the decrease, calculate the difference between the initial total kinetic energy and the final total kinetic energy. This gives the decrease in kinetic energy due to the collision.

Key concepts

Kinetic Energy

Kinetic energy is a form of energy that an object possesses due to its motion. It's calculated using the formula:

• \( KE = \frac{1}{2}mv^2 \)

where \( m \) is the mass, and \( v \) is the velocity of the object. This formula tells us how energy changes when an object's speed or mass changes.
To understand kinetic energy in relation to collisions, consider two people moving on an ice rink. Before and after they collide, we can calculate their kinetic energy based on their mass and speed at those times.
In our example problem, after the collision occurs, you assess whether any energy was lost. In such situations, some kinetic energy is typically converted into other forms due to factors like sound or heat from the impact.
The conservation of momentum does not always imply conservation of kinetic energy, especially in inelastic collisions. This is why calculating both can offer insight into the nature of the collision.

Collision Mechanics

When two objects collide, several physical principles come into play, collectively known as collision mechanics. One important aspect is understanding the difference between elastic and inelastic collisions.
An **elastic collision** is one where both momentum and kinetic energy are conserved. This means no kinetic energy is lost; the objects bounce off each other perfectly.
However, most real-world collisions, such as the one in our frozen pond scenario, tend to be **inelastic**. In these collisions, momentum is still conserved, but some kinetic energy is lost as described earlier.
In our problem, after Sam and Abigail collide, they each move in new directions with new speeds. Calculating their new paths involves breaking down their velocities into components and ensuring that momentum is balanced in all directions.
By comparing initial and final kinetic energies, you can determine how much energy was converted to other forms, offering deeper insights into how the collision unfolded.

Momentum Components

The conservation of momentum principle is vital in solving collision problems and involves considering the initial and final components of momentum in each direction.

• Momentum in the horizontal (east-west) direction.
• Momentum in the vertical (north-south) direction.

The momentum of an object is calculated as the product of its mass and velocity: \( p = mv \).
In our given scenario, Sam and Abigail's collision involves momentum components in two dimensions. Each direction's momentum before the collision must equal the sum of their respective momenta after the collision.
To manage this, breakdown each velocity into components:

• **Eastward component:** Calculated using \( v \times \cos(\theta) \)
• **Northward or Southward component:** Calculated using \( v \times \sin(\theta) \)

By setting up equations for each momentum component, we can solve for the unknowns and understand how their speeds contributed to the overall system momentum. This method allows us to find various desired quantities, like their speeds before and after impact.
It remains crucial for determining how the angle and speed of each engaged in a collision influence their subsequent motion.