Problem 24 - College Physics

Question

\(\bullet\) On a very muddy football field, a \(110-\) kg linebacker tack- les an 85 -kg halfback. Immediately before the collision, the linebacker is slipping with a velocity of 8.8 \(\mathrm{m} / \mathrm{s}\) north and the halfback is sliding with a velocity of 7.2 \(\mathrm{m} / \mathrm{s}\) east. What is the velocity (magnitude and direction) at which the two players move together immediately after the collision?

Step-by-Step Solution

Answer

The final velocity magnitude is approximately 6.52 m/s at 32.4° east of north.

Step 1: Identify Given Values

We are given the masses and velocities of two players involved in a collision. The linebacker's mass is \( m_1 = 110 \, \text{kg} \) and his velocity is \( v_1 = 8.8 \, \text{m/s} \) north. The halfback's mass is \( m_2 = 85 \, \text{kg} \) and his velocity is \( v_2 = 7.2 \, \text{m/s} \) east.

Step 2: Set Up the Conservation of Momentum Equations

According to the law of conservation of momentum, the total momentum before the collision equals the total momentum after the collision. This is expressed as: \[ m_1 \cdot v_1 + m_2 \cdot v_2 = (m_1 + m_2) \cdot v_f \]where \( v_f \) is the final velocity of both players together. Since the velocities are in different directions, we need to consider momentum separately in the north and east directions.

Step 3: Calculate Momentum in North Direction

The momentum in the north direction before the collision is the sum of the linebacker's momentum, as the halfback's momentum is zero in this direction.\[ p_{\text{north}} = m_1 \cdot v_1 = 110 \, \text{kg} \times 8.8 \, \text{m/s} = 968 \, \text{kg} \cdot \text{m/s} \]

Step 4: Calculate Momentum in East Direction

The momentum in the east direction before the collision is the sum of the halfback's momentum, as the linebacker's momentum is zero in this direction.\[ p_{\text{east}} = m_2 \cdot v_2 = 85 \, \text{kg} \times 7.2 \, \text{m/s} = 612 \, \text{kg} \cdot \text{m/s} \]

Step 5: Determine Combined Final Velocity Components

To find the final velocity of the combined mass after the collision, calculate the velocity components in each direction. For north, use:\[ v_{f, \text{north}} = \frac{p_{\text{north}}}{m_1 + m_2} = \frac{968 \, \text{kg} \cdot \text{m/s}}{110 \, \text{kg} + 85 \, \text{kg}} \approx 5.51 \, \text{m/s} \]For east, use:\[ v_{f, \text{east}} = \frac{p_{\text{east}}}{m_1 + m_2} = \frac{612 \, \text{kg} \cdot \text{m/s}}{110 \, \text{kg} + 85 \, \text{kg}} \approx 3.48 \, \text{m/s} \]

Step 6: Calculate the Magnitude of the Final Velocity

Use the Pythagorean theorem to find the magnitude of the combined final velocity.\[ v_f = \sqrt{v_{f, \text{north}}^2 + v_{f, \text{east}}^2} = \sqrt{5.51^2 + 3.48^2} \approx 6.52 \, \text{m/s} \]

Step 7: Determine the Direction of the Final Velocity

Calculate the direction of the final velocity using the arctangent function:\[ \theta = \tan^{-1} \left( \frac{v_{f, \text{east}}}{v_{f, \text{north}}} \right) = \tan^{-1} \left( \frac{3.48}{5.51} \right) \approx 32.4^\circ \]The direction is measured from the north towards the east.

Key Concepts

Momentum Conservation

When two objects collide, the law of momentum conservation comes into play. The total momentum of a system remains constant if no external forces act upon it. This fundamental principle allows us to predict the outcomes of collision scenarios, like the one between the linebacker and the halfback.
In this specific collision, both players become one combined object with a mass equal to the sum of their individual masses. Their velocities contribute to the system's total momentum before and after the collision.

• Before the collision: The linebacker carries momentum north with his motion, and the halfback carries momentum east.
• After the collision: The total momentum is vectors added together to maintain momentum conservation.

To solve for what happens next, we use the idea that the northward and eastward momentum add separately, since they are at right angles to each other.

Vector Components

The concept of vector components is crucial in breaking down complex motion into simpler parts. When objects move in different directions, like this north-moving linebacker and east-moving halfback, we can split their movements into separate vectors.
Vectors help us track how something moves in multiple dimensions. Consider both north and east directions as perpendicular components of the players' movements.

• North component: Only the linebacker has momentum here, expressed as his mass times velocity, because he moves directly north.
• East component: Only the halfback affects this, as his movement is only towards the east.

By understanding these vector components, it simplifies the problem of determining their combined motion after the collision.

Velocity Calculation

Finding the final velocity of the two players moving together involves calculating how these vector components combine. We first solve for each direction separately and then merge the components to get the complete picture.
Here's how it works:

• Determine the northward velocity after the collision: divide the north momentum by the total mass.
• Determine the eastward velocity after the collision: divide the east momentum by the total mass.

These calculations give us the speed in each direction after the collision. They are used to figure out the magnitude and direction of their combined movement.

Pythagorean Theorem

The Pythagorean theorem is not just for geometry but also plays a vital role in collision physics. It's used here to combine the northward and eastward velocity components into a single resultant velocity vector.
Since the north and east velocity components are perpendicular, they form a right triangle. The theorem helps us find the magnitude of the combined velocity:

• Square each component: northward and eastward velocities.
• Add these squares together.
• Take the square root of this sum to find the resultant velocity.

With this method, we determine both the speed and direction at which the two players move together post-collision. Understanding this step ensures a thorough grasp of vector combination and collision outcomes.