Problem 20 - College Physics

Question

An automobile with a linear momentum of \(3.0 \times 10^{4} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\) is brought to a stop in \(5.0 \mathrm{~s}\). What is the magnitude of the average braking force?

Step-by-Step Solution

Answer

The magnitude of the average braking force is \(6.0 \times 10^{3} \, \text{N}\).

Step 1: Understand the Problem

We need to find the magnitude of the average braking force that stops an automobile with an initial linear momentum of \(3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\) over a time of \(5.0 \, \text{s}\).

Step 2: Relate Momentum to Force

Using the impulse-momentum theorem, we can relate force to momentum change: \( F \Delta t = \Delta p \), where \( F \) is the force, \( \Delta t \) is the time interval, and \( \Delta p \) is the change in momentum.

Step 3: Calculate the Change in Momentum

The initial momentum is given as \(3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\) and the final momentum is zero because the car stops. Thus, the change in momentum \( \Delta p = p_f - p_i = 0 - 3.0 \times 10^{4} = -3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\).

Step 4: Solve for the Average Force

Rearrange the impulse-momentum equation to solve for \( F \): \( F = \frac{\Delta p}{\Delta t} \). Substitute \( \Delta p = -3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\) and \( \Delta t = 5 \, \text{s}\) to get \( F = \frac{-3.0 \times 10^{4}}{5} = -6.0 \times 10^{3} \, \text{N}\). The negative sign indicates direction, so the magnitude is \( 6.0 \times 10^{3} \, \text{N}\).

Key Concepts

Average Braking Force

The average braking force is a crucial concept in understanding how forces bring objects to rest. In our case, this refers to the force applied to stop an automobile in motion. According to the impulse-momentum theorem, average braking force (F) is calculated from the change in momentum divided by the time period over which the change occurs.
For instance, if a car has an initial linear momentum of \(3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\) and comes to a stop in \(5 \) seconds, we would find the average braking force by:

• First calculating the change in momentum, which is the initial momentum minus the final momentum (zero, since the car stops).
• Then dividing this change by the time over which it occurred.

The formula \(F = \frac{\Delta p}{\Delta t}\) helps in determining that the average braking force applied in this scenario is \(6.0 \times 10^{3} \, \text{N}\). The absolute value is taken because force as a vector includes direction, and we are often interested just in the magnitude.

Linear Momentum

Linear momentum is an essential quantity in physics, representing the motion of an object. It combines mass and velocity into a single measure, calculated as \(p = m \cdot v\), where \(m\) is mass and \(v\) is velocity.
This concept helps explain how objects move and interact within a system. When an object's momentum changes, it experiences a force. In terms of our automobile example, the initial momentum is given as \(3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\). This value represents the car's quantity of motion at the beginning. When the car is brought to a stop, its velocity becomes zero, hence its momentum reduces to zero.

• Momentum is crucial because it is conserved in isolated systems, meaning that any change in momentum must be the result of external forces.
• Understanding momentum allows for calculating impacts, analyzing motion, and applying forces effectively in everyday physics scenarios.

Change in Momentum

Change in momentum is central to analyzing how forces affect motion. This is calculated as the difference between an object's final and initial momenta, represented as \(\Delta p = p_f - p_i\). Change in momentum ties directly to the concept of impulse, where the impulse imparted by a force over time equals the change in momentum of the object it acts upon.
For the stopped automobile, the initial momentum is \(3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\), and the final momentum is zero as it is brought to rest.
This results in a momentum change of \(-3.0 \times 10^{4} \, \text{kg} \cdot \text{m/s}\). The negative sign indicates the change is in the direction opposite to the initial momentum.

• Understanding change in momentum allows us to determine how much force is required to alter the state of motion of an object.
• This concept is fundamental in collision analysis, safety engineering, and many other areas of physics and engineering.

Identifying and calculating a change in momentum helps in applying the right amount of force for desired outcomes, such as safely stopping a vehicle.