Problem 15 - College Physics

Question

In football practice, two wide receivers run different pass receiving patterns. One with a mass of \(80.0 \mathrm{~kg}\) runs at \(45^{\circ}\) northeast at a speed of \(5.00 \mathrm{~m} / \mathrm{s}\). The second receiver (mass of \(90.0 \mathrm{~kg}\) ) runs straight down the field (due east) at \(6.00 \mathrm{~m} / \mathrm{s}\). (a) What is the direction of their total momentum: (1) exactly northeast, (2) to the north of northeast, (3) exactly east, or (4) to the east of northeast? (b) Justify your answer in part (a) by actually computing their total momentum.

Step-by-Step Solution

Answer

The total momentum direction is to the east of northeast.

Step 1: Calculate Individual Momentums

To determine the total momentum, we first need to calculate the momentum of each receiver. For the first receiver with mass \( m_1 = 80.0 \, \mathrm{kg} \), moving northeast at \( 45^{\circ} \), the velocity components in the east (x) and north (y) directions are \( v_{1x} = 5.00 \, \mathrm{m/s} \times \cos(45^{\circ}) \) and \( v_{1y} = 5.00 \, \mathrm{m/s} \times \sin(45^{\circ}) \). This gives:\[ p_{1x} = m_1 \cdot v_{1x} = 80.0 \, \mathrm{kg} \times 5.00 \, \mathrm{m/s} \times \cos(45^{\circ}), \]\[ p_{1y} = m_1 \cdot v_{1y} = 80.0 \, \mathrm{kg} \times 5.00 \, \mathrm{m/s} \times \sin(45^{\circ}). \]Using \( \cos(45^{\circ}) = \sin(45^{\circ}) = 0.7071 \), we calculate:\[ p_{1x} = 80.0 \times 5.00 \times 0.7071 = 282.84 \, \mathrm{kg \cdot m/s}, \]\[ p_{1y} = 282.84 \, \mathrm{kg \cdot m/s}. \]For the second receiver with mass \( m_2 = 90.0 \, \mathrm{kg} \), running east at \( v_2 = 6.00 \, \mathrm{m/s} \):\[ p_{2x} = m_2 \cdot v_2 = 90.0 \, \mathrm{kg} \times 6.00 \, \mathrm{m/s} = 540.0 \, \mathrm{kg \cdot m/s}. \]The second receiver has no northward component, so \( p_{2y} = 0. \)

Key Concepts

Vector Components of Momentum

In physics, breaking down a vector into components is a fundamental skill when dealing with momentum calculations. Momentum, being a vector quantity, means it has both magnitude and direction.
To understand these components, think of momentum as having two parts: one part pointing east, and another part pointing north. This can be visualized clearly using trigonometry.

• For the receiver moving northeast, we can find the eastward component using the cosine function: \( p_{1x} = m \cdot v \cdot \cos(\theta) \).
• The northward component can be found using the sine function: \( p_{1y} = m \cdot v \cdot \sin(\theta) \).

By using these trigonometric functions, the path splits into two independent directions. This way, you can calculate each piece of momentum separately before adding them together at the end.

Using Trigonometry in Physics

Trigonometry plays a crucial role in many physics problems, especially when it comes to understanding vectors like momentum. It helps to break down complex motions into simpler, more manageable parts.
In this exercise, the use of trigonometry helps us to break down the velocity of an object at a 45-degree angle into horizontal and vertical components.

• At a 45-degree angle, both the sine and cosine functions have the same value, which is approximately 0.7071. This can be a helpful hint when solving problems involving this angle.
• Trigonometry is used to project the velocity onto different axes, which allows us to analyze each dimension independently.

These calculations help us understand how motion in one direction influences another, and are essential for solving problems involving angles.

Collision Analysis and Total Momentum

Collision analysis without a change in mass involves the linear addition of momentum vectors. The objective is to find the combined momentum of two or more objects.

• Momentum is conserved in a closed system. So, the total momentum before and after a collision tends to remain the same.
• By adding the momentum vectors from each player, we can determine the final direction of their combined momentum.

To do this, simply add the components you found earlier. Add the eastward components and the northward components separately:

• Eastward (x-axis): \( p_{total\, x} = p_{1x} + p_{2x} \)
• Northward (y-axis): \( p_{total\, y} = p_{1y} + p_{2y} \)

Finally, the direction of the total momentum can be found using trigonometry again, specifically the tangent function:\[ \text{direction} = \arctan \left( \frac{p_{total\, y}}{p_{total\, x}} \right). \] This angle tells you how much the total momentum deviates from the purely eastward direction. Critical analysis of the speeds and masses involved also provides insights, justifying whether the direction is to the east of northeast or another possible outcome.