Problem 1 - Applied Physics

Question

Two motorcycles of equal mass collide at a \(90^{\circ}\) intersection. If the momentum of motorcycle \(\mathrm{A}\) is \(450 \mathrm{~kg} \mathrm{~km} / \mathrm{h}\) west and the momentum of motorcycle \(\mathrm{B}\) is \(725 \mathrm{~kg}\) \(\mathrm{km} / \mathrm{h}\) south, what is the magnitude of the resulting momentum of the final mass?

Step-by-Step Solution

Answer

The momentum magnitude is approximately 853.9 kg km/h.

Step 1: Understanding the problem

We have two motorcycles colliding at a perpendicular intersection. Motorcycle A is moving west with a momentum of 450 kg km/h, and motorcycle B is moving south with a momentum of 725 kg km/h. We need to find the magnitude of the resulting momentum vector after the collision.

Step 2: Visualizing momentum vectors

Draw a vector diagram representing the momentums: vector A directed west and vector B directed south. The momentum vectors are perpendicular to each other.

Step 3: Applying Pythagorean theorem

Since the vectors are perpendicular, the magnitude of the resultant momentum can be found using the Pythagorean theorem. Let the resultant momentum be denoted as \( R \). You'll calculate \( R \) as follows: \( R = \sqrt{(450)^2 + (725)^2} \).

Step 4: Calculating resultant momentum

Perform the calculations: \( 450^2 = 202500 \) and \( 725^2 = 525625 \). Then, \( R = \sqrt{202500 + 525625} = \sqrt{728125} \).

Step 5: Finding the magnitude

Calculate \( \sqrt{728125} \) to get the magnitude of the resultant momentum: \( R \approx 853.9 \). Thus, the magnitude of the resulting momentum of the final mass is approximately 853.9 kg km/h.

Key Concepts

Vector Analysis

In the context of momentum, vector analysis is a powerful tool used to resolve situations where multiple vectors, such as momentum, are involved. When two objects collide, like the motorcycles in this scenario, their respective momentum vectors must be analyzed to determine their combined effect. Vectors are entities described by both magnitude and direction. For instance, motorcycle A has a momentum vector directed westward and motorcycle B southward. These directions are key in setting up a Cartesian coordinate system where we can visually imagine the vectors.

• Motorcycle A's vector direction is west, represented along the negative x-axis.
• Motorcycle B's direction is south, represented along the negative y-axis.

The vectors form a right-angle since they are perpendicular to each other. To solve the problem, we must consider both magnitude and direction, leading us to utilize the Pythagorean theorem to assess the resultant vector accurately.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry, especially useful when dealing with right triangles. In vector analysis, it helps determine the magnitude of a resulting vector when two vectors are perpendicular (forming a right angle). Given a right-angled triangle, the Pythagorean theorem states:\[ c^2 = a^2 + b^2 \]Where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other sides. In collision physics, this translates to finding the resultant momentum vector's magnitude.For this collision exercise:

• Vector A's magnitude \( (450 \; \text{kg km/h}) \) represents one leg of the triangle.
• Vector B's magnitude \( (725 \; \text{kg km/h}) \) represents the other leg.
• The resultant vector is the hypotenuse.

By applying the theorem:\[ R = \sqrt{(450)^2 + (725)^2} \]This yields the magnitude of the resultant momentum vector. It is a robust technique for resolving scales when directions are at right angles, simplifying calculations and ensuring accuracy.

Collision Physics

In the realm of collision physics, understanding momentum's role is vital. Momentum, a vector quantity, helps determine the outcomes of colliding objects. In this problem with two motorcycles, each with their own momentum, collisions illustrate how they transfer and combine momentum. When dealing with collisions:

• Momentum conservation principle is crucial; the total momentum before and after a collision remains equal, provided there is no external force acting on it.
• Since the motorcycles collide at a right-angle, the result is an easily solvable vector addition without needing to apply external forces.

The principle of momentum conservation and vectors having perpendicular paths leads to using the Pythagorean theorem for resolution. The ultimate outcome is a single vector, the vector sum, which indicates the system's combined momentum post-collision. This principle not only applies to motorcycles but is a core component of understanding traffic collisions, sports physics, and other scenarios where objects physically interact.