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 car's average acceleration is \(0.5\,\text{m/s}^2\) east.

Step-by-step solution

Write the formula for average acceleration

The average acceleration is calculated as follows: $$a_{avg} = \cfrac{v_f - v_0}{\Delta t}$$ Where \(v_0\) is the initial velocity, \(v_f\) is the final velocity, and \(\Delta t\) is the time interval.

Plug in the given values

We have \(v_0=22.0\,\text{m/s}, v_f=17.0\,\text{m/s}\), and \(\Delta t=10.0\,\text{s}\). Let's plug these values into the formula: $$a_{avg} = \cfrac{17.0\,\text{m/s} - 22.0\,\text{m/s}}{10.0\,\text{s}}$$

Calculate the average acceleration

Let's perform the calculation: $$a_{avg} = \cfrac{-5.0\,\text{m/s}}{10.0\,\text{s}} = -0.5\,\text{m/s}^2$$

Determine the direction

Since the result is negative, it means that the acceleration is in the opposite direction of the initial velocity. Since the car is traveling west, the acceleration is directed east. The magnitude of the acceleration is \(0.5\,\text{m/s}^2\), and the direction is east. So, the car's average acceleration is \(0.5\,\text{m/s}^2\) east.

Key concepts

Initial Velocity

Initial velocity, often denoted as v₀, is the speed at which an object begins its movement over a period of time. It's the velocity of the object before it experiences any change in speed due to acceleration or deceleration. In physics problems, understanding the initial velocity is crucial as it serves as the starting point for calculating changes in motion.

For example, if a car moving west at a speed of 22.0 m/s comes to a stop, the initial velocity is the speed prior to any application of brakes or other forces affecting its speed. It is essential to note the direction as well because velocity is a vector quantity, meaning it has both magnitude and direction. When computing changes in motion, both the numerical value and the direction of the initial velocity are taken into account.

Why It Matters

Knowing the initial velocity allows us to determine how the speed and direction of an object changes over time, which is foundational in understanding the object's dynamics under various forces.

Final Velocity

Final velocity, represented as v_f, is the velocity of an object at the end of the time interval considered for its motion. Just like initial velocity, final velocity is a vector, meaning it includes both speed and direction. It's the speed at which an object is moving after it has experienced acceleration or deceleration.

In the context of our example with the car, after applying brakes for 10 seconds, if the car's speed registers as 17.0 m/s in the same westerly direction, this is the final velocity. It's a critical concept for solving problems in physics because it's the resultant velocity from the interplay of starting speed, acceleration, and time.

Comparing Velocities

By comparing the final velocity to the initial velocity, we can determine if the object has accelerated (increased speed) or decelerated (decreased speed), and calculate the rate of this speed change.

Time Interval

Time interval, denoted by Δt, is the duration over which the change in motion takes place. It is essentially the difference between the initial and final time measurements. In motion problems, this interval helps us to understand how quickly changes in velocity occur.

For the car that is initially moving at 22.0 m/s and slows down to 17.0 m/s, the time interval is 10 seconds. This is the period over which the average acceleration is measured. In physics, understanding the time interval is imperative for computing rates of change, like acceleration, which indicates how much an object's speed changes per unit of time.

Time's Role in Motion

The length of the time interval can greatly affect the outcome of a moving object's speed and position, playing a direct role in equations of motion that predict an object's future state.

Acceleration Direction

Acceleration direction informs us about the directional change of an object's velocity. Since acceleration is also a vector, it has both magnitude and direction. The sign of the calculated acceleration value indicates whether the acceleration is in the same direction as the initial motion or in the opposite direction.

In our car example, the acceleration calculation yielded a value of -0.5 m/s², suggesting that the car is slowing down. The negative sign indicates that the acceleration is directed opposite to the initial movement, which in this case is eastward if the initial direction was westward.

Understanding Directional Changes

The direction of acceleration is pivotal in describing how an object's motion changes. If acceleration is in the same direction as the velocity, the object speeds up. Conversely, if it is in the opposite direction, the object slows down. Grasping this concept is key in predicting motion behavior under various force conditions.