Problem 34 - College Physics

Question

A 15000 -N automobile travels at a speed of \(45 \mathrm{~km} / \mathrm{h}\) northward along a street, and a 7500 -N sports car travels at a speed of \(60 \mathrm{~km} / \mathrm{h}\) eastward along an intersecting street. (a) If neither driver brakes and the cars collide at the intersection and lock bumpers, what will the velocity of the cars be immediately after the collision? (b) What percentage of the initial kinetic energy will be lost in the collision?

Step-by-Step Solution

Answer

The final velocity is 10.0 m/s at 56.3° north of east, with 49.1% of the kinetic energy lost.

Step 1: Convert Speeds to m/s

Convert the speeds of both cars from km/h to m/s.1. The automobile: \( 45 \text{ km/h} \times \frac{1000 \text{ m/km}}{3600 \text{ s/h}} = 12.5 \text{ m/s} \)2. The sports car: \( 60 \text{ km/h} \times \frac{1000 \text{ m/km}}{3600 \text{ s/h}} = 16.67 \text{ m/s} \)

Step 2: Calculate Masses from Weights

Determine the mass of each car by dividing their weight by gravity.1. The automobile: \( m_1 = \frac{15000 \text{ N}}{9.8 \text{ m/s}^2} = 1530.61 \text{ kg} \)2. The sports car: \( m_2 = \frac{7500 \text{ N}}{9.8 \text{ m/s}^2} = 765.31 \text{ kg} \)

Step 3: Calculate Initial Momenta

Calculate the momentum of each car before the collision. The momentum is the product of mass and velocity.1. The automobile's momentum \( p_1 \): \( p_1 = m_1 \times v_1 = 1530.61 \times 12.5 = 19132.625 \text{ kg m/s} \) (Northward)2. The sports car's momentum \( p_2 \): \( p_2 = m_2 \times v_2 = 765.31 \times 16.67 = 12753.187 \text{ kg m/s} \) (Eastward)

Step 4: Apply Conservation of Momentum

Use the conservation of momentum to find the combined final velocity of both cars. Since velocities and momenta have directions, consider momenta in northward (y-axis) and eastward (x-axis) directions separately:1. Total mass after the collision: \( m = m_1 + m_2 = 2295.92 \text{ kg} \)2. Momentum conservation in the x-direction (east): \( 0 + p_{2x} = m \cdot v_{fx} \Longrightarrow v_{fx} = \frac{12753.187}{2295.92} = 5.55 \text{ m/s} \)3. Momentum conservation in the y-direction (north): \( p_{1y} + 0 = m \cdot v_{fy} \Longrightarrow v_{fy} = \frac{19132.625}{2295.92} = 8.33 \text{ m/s} \)

Step 5: Determine Magnitude and Direction of Final Velocity

Use the Pythagorean theorem and trigonometry to find the magnitude and angle of the velocity vector:1. Magnitude: \( v_f = \sqrt{v_{fx}^2 + v_{fy}^2} = \sqrt{(5.55)^2 + (8.33)^2} = 10.0 \text{ m/s} \)2. Angle: \( \theta = \tan^{-1} \left( \frac{v_{fy}}{v_{fx}} \right) = \tan^{-1} \left( \frac{8.33}{5.55} \right) \approx 56.3^\circ \) north of east.

Step 6: Calculate Initial Kinetic Energy

Calculate the initial kinetic energies of both vehicles before the collision:1. Kinetic energy of the automobile: \( KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \times 1530.61 \times (12.5)^2 = 119442.58 \text{ J} \)2. Kinetic energy of the sports car: \( KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \times 765.31 \times (16.67)^2 = 105972.28 \text{ J} \)3. Total initial kinetic energy: \( KE_{initial} = 119442.58 + 105972.28 = 225414.86 \text{ J} \)

Step 7: Calculate Final Kinetic Energy

Determine the kinetic energy after the collision:1. \( KE_{final} = \frac{1}{2} m v_f^2 = \frac{1}{2} \times 2295.92 \times (10)^2 = 114796 \text{ J} \)

Step 8: Calculate Percentage of Energy Lost

Find the percentage of kinetic energy lost during the collision:1. Energy lost: \( = KE_{initial} - KE_{final} = 225414.86 - 114796 = 110618.86 \text{ J} \)2. Percentage lost: \( \frac{110618.86}{225414.86} \times 100 \approx 49.1\% \)

Key Concepts

Momentum Conservation

Momentum conservation is a fundamental principle in physics that describes how momentum is maintained in an isolated system unless acted on by external forces. When two objects collide, such as the automobile and sports car in our problem, the total momentum before the collision is equal to the total momentum after the collision.
It's important to consider momentum as a vector, as it has both magnitude and direction. In our scenario:

• The automobile, moving northward with a known velocity and mass, contributes to the system's momentum in the y-direction (north-south).
• The sports car, traveling eastward, provides momentum in the x-direction (east-west).

To solve the problem, we review each component separately, applying the principle of momentum conservation in both directions. This approach lets us compute the velocities after the collision accurately.
Ultimately, momentum conservation ensures that despite changes in speed and direction post-collision, the combined mass center tends to move inertly, according to the pre-collision total momentum.

Kinetic Energy

Kinetic energy reflects the energy an object possesses due to its motion. In collision problems, analyzing kinetic energy helps us understand the nature of the interaction, notably whether it is elastic or inelastic.
In the scenario given, both cars have significant initial kinetic energies, calculated with the formula:

• Initial kinetic energy: \( KE = \frac{1}{2}mv^2 \)
• Each car's mass and velocity compute to their respective kinetic energies.

During an inelastic collision like this one, where the cars lock bumpers, a portion of kinetic energy transforms, usually into heat or sound, altering the system's total kinetic energy.
The difference in kinetic energy before and after the collision allows us to compute energy lost, giving us insight into how "inefficient" the collision is. In our problem, we're able to determine that nearly half of the initial kinetic energy is dissipated through the collision processes.

Vector Analysis

Vector analysis is a crucial tool when dealing with collision mechanics, especially since forces and velocities in physics are vectors, having both directions and magnitudes.
In our exercise, it's essential to decompose the velocity vectors into their x and y components so that momentum conservation can be applied effectively in each direction.

• The automobile's velocity is purely northward, affecting the y-direction only.
• The sports car's velocity affects the x-direction as it heads east.

Using vector addition, we determine the resultant velocity vector of the combined auto-sports car system post-collision by:

• Finding the x and y components through momentum conservation.
• Computing the magnitude of the resultant vector using the Pythagorean theorem.

This method not only gives us the final speed but also the direction, illustrating the power of vector analysis in breaking down complex movements and changes post-collision.

Physics Problem Solving

Solving physics problems, like this collision scenario, requires a structured approach to untangle complexities and reach meaningful solutions.
First, we translate real-world quantities (like vehicle speeds in km/h) to suitable units (m/s) for physics calculations. Afterward, we follow these steps:

• Determine masses from weights using gravitational acceleration.
• Calculate initial momenta and kinetic energies, essential for analyzing the collision.
• Apply conservation laws to find resultant velocities and energy transformations.

This step-by-step problem-solving approach helps ensure we handle vector components correctly, apply physical laws accurately, and extract meaningful insights—such as velocity post-collision or kinetic energy lost.
This structured method helps in simplifying and solving real-world physics problems with precision and clarity.