Problem 3 - Applied Physics

Question

Two vehicles collide at a \(90^{\circ}\) intersection. If the momentum of vehicle \(\mathrm{A}\) is \(6.10 \times 10^{5} \mathrm{~kg} \mathrm{~km} / \mathrm{h}\) south and the momentum of vehicle \(\mathrm{B}\) is \(7.20 \times 10^{5} \mathrm{~kg} \mathrm{~km} / \mathrm{h}\) east, what is the magnitude of the resulting momentum of the final mass?

Step-by-Step Solution

Answer

The resultant momentum magnitude is \(9.44 \times 10^{5} \, \text{kg} \, \text{km/h}\).

Step 1: Identify the Components of the Problem

We are given two perpendicular momenta: Vehicle A has a momentum of \(6.10 \times 10^{5} \, \text{kg} \, \text{km/h}\) south, and Vehicle B has a momentum of \(7.20 \times 10^{5} \, \text{kg} \, \text{km/h}\) east. To find the resultant momentum, we will consider the components along south and east directions.

Step 2: Apply the Pythagorean Theorem

Since momentum vectors are at \(90^\circ\) to each other, use the Pythagorean theorem to calculate the magnitude of the resultant momentum: \[P_\text{resultant} = \sqrt{(P_\text{A})^2 + (P_\text{B})^2}\]Here, \(P_\text{A} = 6.10 \times 10^{5} \, \text{kg} \, \text{km/h}\) and \(P_\text{B} = 7.20 \times 10^{5} \, \text{kg} \, \text{km/h}\).

Step 3: Substitute Values into the Formula

Substitute the given momentum values into the Pythagorean equation: \[P_\text{resultant} = \sqrt{(6.10 \times 10^{5})^2 + (7.20 \times 10^{5})^2}\]

Step 4: Calculate Each Term in the Equation

First, calculate the square of each momentum:- \((6.10 \times 10^{5})^2 = 3.721 \times 10^{11} \, \text{kg}^2 \, \text{km}^2/\text{h}^2\)- \((7.20 \times 10^{5})^2 = 5.184 \times 10^{11} \, \text{kg}^2 \, \text{km}^2/\text{h}^2\)

Step 5: Add the Squared Terms

Add the squared terms to obtain the sum:\[3.721 \times 10^{11} + 5.184 \times 10^{11} = 8.905 \times 10^{11} \, \text{kg}^2 \, \text{km}^2/\text{h}^2\]

Step 6: Calculate the Resultant Magnitude

Now take the square root of the sum to find the magnitude of the resultant momentum:\[P_\text{resultant} = \sqrt{8.905 \times 10^{11}} = 9.44 \times 10^{5} \, \text{kg} \, \text{km/h}\]

Key Concepts

Vectors in Momentum

Momentum is a vector quantity, meaning it has both magnitude and direction. In this problem, we consider two momentum vectors: one acting toward the south and the other toward the east. Vectors are often represented using arrows, with the length of the arrow indicating magnitude and the direction of the arrow showing direction.
These vectors help in understanding and visualizing the physical situation. Since vehicle A's momentum points south, and vehicle B's points east, they form a right angle. This perpendicular relationship makes calculations straightforward when finding the total momentum.
Working with vectors requires us to analyze each component independently, making them crucial in physics, especially in collision scenarios where different forces interact.

Pythagorean Theorem in Physics

The Pythagorean Theorem is a mathematical principle used to calculate the resultant of two perpendicular vectors. It states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In mathematical terms, this is expressed as \( c^2 = a^2 + b^2 \).
In our exercise, vehicle A's momentum and vehicle B's momentum act like legs of a right triangle, where the hypotenuse is the resultant momentum. By substituting the known values into the Pythagorean formula, we effectively solve for the total momentum of the system after the collision. This theorem is not only a cornerstone in geometry but also a powerful tool in physics for solving real-world problems.

Resultant Momentum Calculation

Resultant momentum is the vector sum of all individual momenta. It is essential in determining the effect of collisions as it helps in predicting the direction and magnitude of a system's movement post-collision.
In our scenario, to find the resultant momentum, we take the square root of the sum of the squares of the individual momenta: \( P_\text{resultant} = \sqrt{(P_\text{A})^2 + (P_\text{B})^2} \).
This formula results from applying the Pythagorean Theorem, and it lets us understand how different forces combine. The calculated resultant momentum indicates how the two colliding vehicles will behave after impact, pointing to a new direction that's neither purely east nor purely south.

Collision Physics and Momentum Conservation

Collision physics deals with the motion of objects before, during, and after collisions. Two main types of collisions are studied: elastic, where kinetic energy is conserved, and inelastic, where it is not. Regardless of the type, the law of conservation of momentum always holds.
In the exercise, the '90-degree collision' results in a direct application of momentum conservation principles. Since momentum is conserved in any closed system, the total momentum before the collision is equal to the total momentum after the collision, assuming no external forces.
Through studying collisions, we gain insights into vehicle safety, sports, and even astronomical events, making it a vital area of physics. Understanding collision dynamics helps in designing safer cars and predicting the outcomes of similar real-world situations.