Question

A car is traveling west at \(22.0 \mathrm{~m} / \mathrm{s}\). After \(10.0 \mathrm{~s}\), its velocity is \(17.0 \mathrm{~m} / \mathrm{s}\) in the same direction. Find the magnitude and direction of the car's average acceleration.

Short answer

Answer: The magnitude of the car's average acceleration is 0.5 m/s², and the direction is East.

Step-by-step solution

Identify the given information

In this problem, we are given the following information: - Initial velocity of the car (\(v_i\)) = 22.0 m/s (west) - Final velocity of the car (\(v_f\)) = 17.0 m/s (west) - Time duration (\(\Delta t\)) = 10.0 s

Calculate the change in velocity

To calculate the change in velocity (\(\Delta v\)), we subtract the initial velocity from the final velocity. Since both velocities have the same direction, we can simplify this as follows: \(\Delta v = v_f - v_i = 17.0 - 22.0 = -5.0 \mathrm{~m} / \mathrm{s}\) The negative sign indicates that the car is slowing down in the westward direction.

Calculate the average acceleration

Now, we can use the formula for average acceleration to find the magnitude and direction of the car's average acceleration. The formula is: \( a_{avg} = \frac{\Delta v}{\Delta t}\) By plugging in the given values, we get: \(a_{avg} = \frac{-5.0}{10.0} = -0.5 \mathrm{~m} / \mathrm{s^2}\) The negative sign indicates that the car's acceleration is in the opposite direction of its velocity, which means it is slowing down.

Determine the direction of the average acceleration

Since the car is slowing down while traveling west, the direction of the average acceleration is to the east.

Write down the magnitude and direction of the average acceleration

Finally, the magnitude and direction of the car's average acceleration is: Magnitude: 0.5 m/s² Direction: East

Key concepts

Initial and Final Velocity

Understanding the concepts of initial and final velocity is crucial in kinematics, the branch of physics that describes the motion of objects. When an object starts moving, it has an initial velocity, denoted as \( v_i \). This is the velocity of the object at the beginning of the time interval in question. As the object moves and time progresses, it may speed up, slow down, or change direction, leading to a final velocity, \( v_f \), which is the velocity at the end of the time interval.

When analyzing the motion of a car that initially travels west at \(22.0 \text{ m/s}\) and later has a final velocity of \(17.0 \text{ m/s}\) in the same direction, we are observing a decrease in speed. The car is still traveling west, but its speed at the end of the 10-second interval is slower than when it started. This change of speed over time is intricately connected to the concept of acceleration, which will be discussed in the following sections.

Change in Velocity

The change in velocity is a vector quantity that describes the difference between an object's final velocity and its initial velocity, symbolized as \( \triangle v \). Calculating this change allows us to understand how the object's motion has evolved over a certain period. It is represented by the equation:\[ \triangle v = v_f - v_i \]

In the case of the car moving west, the initial velocity is \(22.0 \text{ m/s}\) and the final velocity after 10 seconds is \(17.0 \text{ m/s}\). By subtracting the initial velocity from the final velocity, we get a negative result of \(-5.0 \text{ m/s}\). This negative sign is significant as it indicates the car is decelerating, or in other words, its speed is decreasing over time while it maintains its direction westward.

Direction of Acceleration

The direction of acceleration is a fundamental aspect when discussing changes in velocity and kinematics as a whole. Acceleration doesn't merely refer to increases in speed; it applies to any change in velocity, which includes slowing down and changes in direction. If the acceleration vector points in the same direction as the velocity vector, the object is speeding up. Conversely, if the acceleration vector is in the opposite direction, the object is slowing down.

For the westward-moving car, a negative acceleration means the car's acceleration is eastward. It is decelerating because the acceleration is opposite to the initial velocity's direction. In kinematics, this principle helps us understand that the object's direction of travel isn't merely a matter of where it's moving towards but also how its speed is being influenced by its acceleration.

Kinematics

The study of kinematics revolves around describing motion without considering the forces that cause it. It deals with concepts like velocity, acceleration, displacement, and time. Kinematics enables us to solve problems related to the motion of objects by using equations and known values.

Equations of kinematics often include terms for initial and final velocities, acceleration, time, and displacement. In our example of the car, kinematic principles allow us to deduce the car’s average acceleration during the 10-second interval. With values for initial and final velocities and the time period, we can calculate that the car experiences an average acceleration of \(-0.5 \text{ m/s}^2\), which tells us not only the magnitude of the car's rate of change in speed but also the direction of this change. Understanding kinematics is essential for predicting future motion based on current and past motion states.