Problem 11 - Applied Physics

Question

A \(1650-\mathrm{kg}\) automobile moving south \(12.0 \mathrm{~m} / \mathrm{s}\) collides with a \(2450-\mathrm{kg}\) automobile moving north on an icy road. The automobiles stick together and move \(3.00 \mathrm{~m} / \mathrm{s}\) to the north after the collision. What is the speed of the heavier automobile before the collision?

Step-by-Step Solution

Answer

The speed of the heavier automobile before the collision was 13.1 m/s.

Step 1: Understand the Problem

We have a collision between two automobiles. After colliding, they move together with a specific velocity. We need to find the speed of the heavier automobile before the collision.

Step 2: Apply Conservation of Momentum

In a closed system, momentum before collision equals momentum after collision. Here, each automobile's momentum before the collision should add up to the combined momentum after the collision.

Step 3: Calculate Initial Momentum

The momentum of the first automobile is given by its mass times its velocity: \( p_1 = 1650 \, \text{kg} \times (-12.0 \, \text{m/s}) = -19800 \, \text{kg} \cdot \text{m/s} \) (negative because it's moving south). The second car's initial momentum is \( p_2 = 2450 \, \text{kg} \times v_2 \), where \( v_2 \) is what we want to find.

Step 4: Calculate Final Momentum

After the collision, they stick together: \( (1650 \, \text{kg} + 2450 \, \text{kg}) \cdot 3.00 \, \text{m/s} = 12300 \, \text{kg} \cdot \text{m/s} \) (as they move north, hence positive).

Step 5: Set Up the Equation

Using the conservation of momentum: \( -19800 \, \text{kg} \cdot \text{m/s} + 2450 \, \text{kg} \cdot v_2 = 12300 \, \text{kg} \cdot \text{m/s} \).

Step 6: Solve for \( v_2 \)

Rearrange the equation to solve for \( v_2 \): \[ 2450 \, \text{kg} \cdot v_2 = 12300 \, \text{kg} \cdot \text{m/s} + 19800 \, \text{kg} \cdot \text{m/s} \] \[ v_2 \times 2450 = 32100 \] \[ v_2 = \frac{32100}{2450} = 13.1 \, \text{m/s} \].

Key Concepts

Collision Physics

Collision physics involves the study of how objects interact upon impact. In this exercise, we are dealing with a **perfectly inelastic collision**. This means that after the automobiles collide, they move together as one mass. Such collisions are characterized by maximum kinetic energy loss, which transforms into other forms of energy, like sound or heat, and sometimes gets absorbed by the materials involved.

There are different types of collisions, but inelastic collisions do not conserve kinetic energy. It's essential to understand that while the kinetic energy may not be conserved, **momentum is always conserved** in a closed system like this one. This guiding principle allows us to solve for unknown variables, such as the velocity of one of the cars before the collision, as demonstrated in the steps provided. By applying the conservation of momentum, we can derive the initial conditions of motion before the collision.

Momentum Calculation

Momentum is a fundamental concept in physics, defined as the product of an object's mass and velocity. Represented by the symbol \( p \), it is mathematically expressed as:

• \( p = m \cdot v \)

For this exercise, we calculated the momentum of each car before the collision. The momentum of the first car was straightforward as its mass and velocity were provided.
The momentum formula used was:

• Car 1: \(-19800 \, \text{kg} \cdot \text{m/s}\)
• Car 2: \(2450 \, \text{kg} \cdot v_2\)

where the negative sign indicated the direction of motion (southward).

Calculating momentum gives you an idea of how strongly motion is being carried in a certain direction. In collisions like these, understanding momentum helps determine how objects will move post-collision. Both vehicles' momenta sum up to give the combined momentum, allowing us to backtrack and discover unknown initial velocities.

Velocity Determination

Finding velocity can be a challenging task, particularly in collision scenarios where only some data points are known. However, with momentum conservation at our disposal, it becomes manageable. In this problem, we aimed to determine the initial velocity of the heavier car:

• We first calculated the initial momentum of the lighter car moving south.
• The total momentum post-collision was determined to be positive because the cars moved north.

By setting up the **momentum conservation equation** for the collision, we equated the total initial momentum to the final momentum:

• \(-19800 \, \text{kg} \cdot \text{m/s} + 2450 \, \text{kg} \cdot v_2 = 12300 \, \text{kg} \cdot \text{m/s}\)

This equation was then solved for \( v_2 \), the heavier automobile's velocity before the collision, resulting in \( v_2 = 13.1 \, \text{m/s}\).
Applying this approach of setting up equations helps tackle these physics problems systematically.