Problem 23 - College Physics

Question

On a very muddy football field, a \(110 \mathrm{~kg}\) linebacker tackles an \(85 \mathrm{~kg}\) 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 velocity is 5.87 m/s at 32.5° east of north.

Step 1: Understand the Conservation of Momentum

We will use the principle of conservation of momentum, which states that the total momentum before collision equals the total momentum after collision, as no external forces are acting on the system. The formula for momentum is \( p = m \cdot v \), where \( m \) is mass and \( v \) is velocity.

Step 2: Calculate the Initial Momentum of Each Player

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

Step 3: Calculate the Total Initial Momentum

Since the football field directions are north and east, treat this as a two-dimensional vector problem. The total initial momentum vector \( \mathbf{p}_{initial} \) is:- North component: 968 kg m/s- East component: 612 kg m/s

Step 4: Find the Combined Mass of the Players

The total mass after collision is the sum of both players' masses: \( m_{total} = 110 \text{ kg} + 85 \text{ kg} = 195 \text{ kg} \).

Step 5: Calculate the Final Velocity Magnitude

Using the momentum principle, the final velocity in each direction can be found by dividing the momentum by the total mass. Find the north velocity component: \( v_{n} = \frac{968}{195} = 4.964 \text{ m/s} \).Find the east velocity component: \( v_{e} = \frac{612}{195} = 3.138 \text{ m/s} \).Calculate the magnitude of the final velocity using Pythagorean theorem: \[ v = \sqrt{v_{n}^{2} + v_{e}^{2}} = \sqrt{4.964^{2} + 3.138^{2}} = 5.87 \text{ m/s} \].

Step 6: Calculate the Direction of the Final Velocity

Find the angle \( \theta \) using the tangent function: \( \tan \theta = \frac{v_{e}}{v_{n}} = \frac{3.138}{4.964} \).Calculate \( \theta \) using the arctangent function: \( \theta = \tan^{-1}\left(\frac{3.138}{4.964}\right) = 32.5° \).The direction is 32.5° east of north.

Key Concepts

Two-Dimensional Vector

When dealing with physics problems, it's important to understand how two-dimensional vectors work. A vector is a quantity that has both magnitude and direction. In the given problem, we have two vectors to handle: the linear momentum of each football player, which is a vector quantity.
These vectors are two-dimensional because they involve both the north-south axis and the east-west axis on the field:

• The linebacker moves north with a certain momentum.
• The halfback moves east with another momentum.

In two-dimensional vector analysis, each vector can be broken down into components along the chosen axes—in this case, north and east. Understanding how these vectors combine in two dimensions helps us determine how the two players' movements result in motion along a single resultant direction after the collision.

Collisions

Collisions are events where two or more bodies exert forces on each other for a relatively short duration. Here, we are dealing with a perfectly inelastic collision, where the two players stick together after colliding.
In such collisions, the conservation of momentum principle plays a crucial role: the total momentum before the collision is equal to the total momentum after the collision.
This principle allows us to predict the state of the motion post-collision, helping us find the final velocity of the two players as they move together. The motion can be visualized as a combination of their original speeds, creating a new path of motion determined by their collective mass and momentum.

Momentum Calculation

Momentum is a measure of the motion of a body and is given by the product of mass and velocity, denoted as \( p = m \cdot v \). Calculating momentum is foundational to solving collision problems.
Initially, the momentum of each player is calculated separately:

• The linebacker: \( p_l = 110 \text{ kg} \times 8.8 \text{ m/s} = 968 \text{ kg m/s north} \)
• The halfback: \( p_h = 85 \text{ kg} \times 7.2 \text{ m/s} = 612 \text{ kg m/s east} \)

The key to solving the problem lies in using these momentum values correctly to find the resulting momentum of the system (both players together). This is a vector sum of the momentum components and is calculated by combining the north and east momenta.

Vector Components

In any two-dimensional motion, breaking vectors into components helps simplify calculations. A vector component is the projection of a vector along the axes of a coordinate system. Here, the system uses north and east directions as axes.
For the momentum vectors, we need to:

• Identify the north component, contributed entirely by the linebacker: 968 kg m/s.
• Identify the east component, contributed entirely by the halfback: 612 kg m/s.

The final velocity of the two players post-collision is derived by calculating these components' resultant. For the north and east components of the final velocity, we divide the respective momentum by the total mass, then use the Pythagorean theorem to find the magnitude.
Lastly, the direction is found using trigonometric functions, providing a full picture of the movement direction in terms of angle.