Problem 37 - University Physics with Modern Physics

Question

On a very muddy football field, a 110 -kg linebacker tackles an \(85-\) kg halfback. Immediately before the collision, the line-backer 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 players move at 5.91 m/s, 32.04° east of north.

Step 1: Understand the Law of Conservation of Momentum

The principle of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision. Since they move together after the collision, they form a single object with combined mass.

Step 2: Calculate Initial Momentum of Each Player

Calculate the momentum of the linebacker: \( p_{\text{linebacker}} = m_{\text{linebacker}} \times v_{\text{linebacker}} = 110 \times 8.8 = 968 \ \mathrm{kg}\cdot\mathrm{m/s} \) north. Calculate the momentum of the halfback: \( p_{\text{halfback}} = m_{\text{halfback}} \times v_{\text{halfback}} = 85 \times 7.2 = 612 \ \mathrm{kg}\cdot\mathrm{m/s} \) east.

Step 3: Determine the Combined Mass

Add the masses of both players to find the combined mass after the collision: \( m_{\text{total}} = 110 + 85 = 195 \ \mathrm{kg} \).

Step 4: Find the Total Momentum Components

Calculate the total momentum in the north and east directions. North: \( p_{\text{total}}^{\text{north}} = 968 \ \mathrm{kg}\cdot\mathrm{m/s} \). East: \( p_{\text{total}}^{\text{east}} = 612 \ \mathrm{kg}\cdot\mathrm{m/s} \).

Step 5: Calculate the Resultant Momentum

Use the Pythagorean theorem to find the magnitude of the resultant momentum: \( p_{\text{resultant}} = \sqrt{(968)^2 + (612)^2} \approx 1152.2 \ \mathrm{kg}\cdot\mathrm{m/s} \).

Step 6: Calculate the Velocity After Collision

Using the resultant momentum and total mass, calculate the velocity: \( v = \frac{p_{\text{resultant}}}{m_{\text{total}}} = \frac{1152.2}{195} \approx 5.91 \mathrm{m/s} \).

Step 7: Find the Direction of the Movement

Use trigonometry to find the angle: \( \theta = \tan^{-1}\left(\frac{612}{968}\right) \approx 32.04^{\circ} \) east of north.

Key Concepts

Momentum Calculation

The momentum of an object is a key concept in physics, referring to the product of its mass and velocity. This quantity describes the amount of motion an object has and is essential in analyzing collisions. In our given exercise, we calculate the momentum of two football players before they collide.
The linebacker, weighing 110 kg, with a velocity of 8.8 m/s north, has a momentum of 968 kg·m/s. Meanwhile, the halfback, weighing 85 kg and moving east at 7.2 m/s, has a momentum of 612 kg·m/s.
Key points to remember about momentum calculation:

• Momentum is a vector quantity. This means it has both magnitude and direction.
• The units of momentum are expressed in kg·m/s.
• Total momentum in a system remains constant if no external forces act on it.

The calculation of momentum involves straightforward multiplication of mass and velocity, but keep in mind that direction plays a crucial role, especially in understanding collisions.

Collision Physics

In collision physics, particularly elastic and inelastic collisions, conservation of momentum plays a significant role. In the exercise, we're dealing with an inelastic collision where two bodies stick together post-collision. Once two objects collide, they form a single combined mass that moves with a new, common velocity.
The principle guiding collisions is the conservation of momentum. Before the collision, each player carries a portion of the total system momentum. The post-collision velocity is determined by distributing the total momentum over the combined mass of the objects.

• Inelastic collisions like the football example conserve momentum but not necessarily kinetic energy.
• The calculation of post-collision velocity uses the total momentum and the combined mass.
• Understanding real-world deformations during collisions helps explain why collisions may not conserve kinetic energy.

By grasping conservation laws, students can predict the result of any collision with or without additional forces involved.

Vector Addition

Vector addition is crucial in solving problems involving multiple directional components, like in our collision problem. Players moving in perpendicular directions represent north and east momentum components, which require vector addition to achieve a resultant direction and magnitude.
The resultant vector is calculated using the Pythagorean theorem by combining individual momentum components: north and east. In our exercise, the resultant momentum (\[ p_{\text{resultant}} = \sqrt{(968)^2 + (612)^2} \approx 1152.2 \ \mathrm{kg\cdot m/s} \]) represents the total combined motion post-collision.

• Vectors must be added geometrically by considering both x and y components.
• The magnitude gives the combined effect's size, while the direction is determined using trigonometric functions.
• Understanding the parallelogram or triangle method of vector addition can simplify solving problems with right-angle components.

Mastering vector addition is crucial for anyone delving into physics, as it applies across various topics beyond just momentum and collisions.