Problem 13 - College Physics

Question

A car is stopped for a traffic signal. When the light turns green, the car accelerates, increasing its speed from 0 to \(5.20 \mathrm{~m} / \mathrm{s}\) in \(0.832 \mathrm{~s}\). What are (a) the magnitudes of the linear impulse and (b) the average total force experienced by a \(70.0-\mathrm{kg}\) passenger in the car during the time the car accelerates?

Step-by-Step Solution

Answer

The magnitude of the linear impulse is \(364 \mathrm{~kg \cdot m/s}\) and the average total force experienced by the passenger is \(437.5 \mathrm{~N}\)

Step 1: Calculate Linear Impulse

Impulse can be calculated using the formula Impulse = mass * change in velocity. The change in velocity is the final velocity minus the initial velocity. Here, the final velocity of the car is \(5.20 \mathrm{~m/s}\) and the initial velocity is \(0 \mathrm{~m/s}\), so the change in velocity is \(5.20 \mathrm{~m/s}\). The mass of the passenger is \(70.0 \mathrm{~kg}\), so the impulse is \(70.0 \mathrm{~kg} * 5.20 \mathrm{~m/s} = 364 \mathrm{~kg \cdot m/s}\)

Step 2: Calculate Acceleration

Acceleration is the change in velocity divided by the change in time. From the previous step we know that the change in velocity is \(5.20 \mathrm{~m/s}\). The time it takes for the car to accelerate is given as \(0.832 \mathrm{~s}\), so the acceleration is \(5.20 \mathrm{~m/s} / 0.832 \mathrm{~s} = 6.25 \mathrm{~m/s^2}\)

Step 3: Calculate Average Total Force

Force is mass times acceleration. From the previous steps we know that the mass of the passenger is \(70.0 \mathrm{~kg}\) and the acceleration of the car is \(6.25 \mathrm{~m/s^2}\). Therefore, the force experienced by the passenger is \(70.0 \mathrm{~kg} * 6.25 \mathrm{~m/s^2} = 437.5 \mathrm{~N}\)

Key Concepts

Impulse Formula

Understanding the impulse formula is crucial in physics, as it helps describe how the momentum of an object changes when it's subjected to a force over a period of time. Impulse is defined as the product of the average force applied to an object and the time duration over which the force is applied. This is mathematically represented as:

\[ \text{Impulse} = \text{Force} \times \text{Time} \]
or, since force is mass times acceleration, \[ \text{Impulse} = \text{Mass} \times \Delta\text{Velocity} \]
where \( \Delta\text{Velocity} \) is the change in velocity.

In the case of our exercise, where a passenger in a car experiences a change in velocity from rest to \(5.20 \text{ m/s}\), the impulse can be easily calculated by multiplying the mass of the passenger by this change in velocity. The formula captures the idea that if you push something for a longer time, or with more force, the total impulse delivered, and hence the object's change in momentum, will be greater. This understanding of impulse is essential in analyzing various real-world scenarios, such as car crashes, jumps in sports, or the firing of projectiles.

Acceleration Calculation

Acceleration represents how quickly an object speeds up, slows down, or changes direction. It is a vector quantity, meaning it has both magnitude and direction. The standard formula to calculate linear acceleration is:

\[ \text{Acceleration} = \frac{\Delta\text{Velocity}}{\Delta\text{Time}} \]
where \( \Delta\text{Velocity} \) is the change in velocity and \( \Delta\text{Time} \) is the time period over which this change occurs.

By applying this formula to our exercise, where a car accelerates over 0.832 seconds, we can find the acceleration by dividing the velocity change of \(5.20 \text{ m/s}\) by the time interval. Precise calculation of acceleration is vital as it is used in numerous physics equations and principles, including determining the forces acting on moving objects as seen in our application of the concept to find the average total force on a passenger.

Newton's Second Law

Newton's Second Law of Motion is a cornerstone of classical mechanics and states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. This is commonly written as:

\[ \text{Force} = \text{Mass} \times \text{Acceleration} \]
According to this principle, if you know the mass of an object and its acceleration, you can calculate the force. In our exercise, applying Newton's Second Law helps us find the average total force exerted on the passenger as the car speeds up.

Understanding Newton's Second Law is fundamental to predicting how objects will move under the influence of various forces. It ties into the impulse formula as well, since force can also be understood as the change in momentum (impulse) per unit of time. Consequently, this law provides the framework for analyzing motion and is a critical concept in not only physics but also in engineering, aeronautics, and various other scientific and practical fields.