Question

A car starting at rest accelerates at \(16 \mathrm{ft} / \mathrm{s}^{2}\) for 5 seconds on a straight road. How far does it travel during this time?

Short answer

Answer: The car travels a distance of 200 feet during the 5 seconds of acceleration.

Step-by-step solution

Identify the given variables

We are given the following information: - Initial velocity \((v_0)\) is 0 (since the car starts at rest) - Acceleration \((a)\) is \(16 \mathrm{ft} / \mathrm{s}^{2}\) - Time \((t)\) is 5 seconds

Calculate the final velocity

To find the final velocity \((v)\), we'll use the following kinematic equation that relates initial velocity, final velocity, acceleration, and time: \(v = v_0 + a t\) Since \(v_0 = 0\) and we have all other values, we can plug them into the formula: \(v = 0 + (16 \ \mathrm{ft} / \mathrm{s}^{2})(5 \ \mathrm{s})\) \(v = 80 \ \mathrm{ft} / \mathrm{s}\)

Calculate the total distance

To find the distance or displacement \((d)\) traveled during this time, we can use another kinematic equation that relates initial velocity, final velocity, acceleration, time, and distance: \(d = v_0 t + 1/2 a t^2\) Again, since \(v_0 = 0\), the formula simplifies to: \(d = 1/2 a t^2\) Now, we can plug in the values for acceleration and time: \(d = 1/2 (16 \ \mathrm{ft} / \mathrm{s}^{2})(5 \ \mathrm{s})^2\) \(d = 1/2 (16 \ \mathrm{ft} / \mathrm{s}^{2})(25 \ \mathrm{s}^2)\) \(d = 200 \ \mathrm{ft}\) Therefore, the car travels a distance of 200 feet during the 5 seconds of acceleration.

Key concepts

Understanding Initial Velocity

Initial velocity is the speed at which an object begins its motion. It's important for calculating how motion evolves. In the given problem, the car starts from rest, so the initial velocity \(v_0\) is 0. This zero initial velocity simplifies calculations, allowing focus on acceleration's impact. When initial velocity is zero, it simplifies many kinematic equations. These equations are crucial in predicting how an object will move given specific conditions.

In practical scenarios, the initial velocity can be anything but zero, depending on the situation, such as a car moving before accelerating. Understanding initial velocity helps comprehend how motion can vary based on starting conditions.

Exploring Final Velocity

Final velocity refers to the speed of an object at the end of a given period of time or after certain influences have acted upon it. In kinematics, it's crucial to determine the changes an object undergoes during its motion. For the car problem, the final velocity \(v\) is calculated using the equation \(v = v_0 + a t\).

Given that the car was initially at rest, the equation simplifies to \(v = 80 \, \mathrm{ft/s}\) after acceleration. This final velocity tells us how quickly the car is moving at the end of the 5-second period and provides insight into the car's motion. Calculating final velocity is fundamental in scenarios like determining stopping distances or predicting motion paths.

Deciphering Acceleration

Acceleration is the rate at which an object's velocity changes over time. In kinematics, it describes how quickly an object speeds up or slows down. For the car in the problem, the acceleration \(a\) is given as \(16 \, \mathrm{ft/s^2}\). This means that every second, the car's velocity increases by 16 feet per second.

In the context of the problem, acceleration allows us to calculate both the final velocity and the distance traveled. By using equations like \(d = \frac{1}{2} a t^2\), we can see the effect of this steady acceleration over a period of time. In real-world scenarios, knowing an object's acceleration can help in planning safe travel, assessing performance, and understanding dynamics.