Question

An acceleration function of an object moving along a straight line is given. Find the change of the object's velocity over the given time interval. $$ a(t)=t \mathrm{ft} / \mathrm{s}^{2} \text { on }[0,2] $$

Short answer

The change in velocity is 2 ft/s.

Step-by-step solution

Understand the Problem

We are given the acceleration function \(a(t) = t\) in \(\text{ft/s}^2\), and we need to find the change in velocity over the time interval \([0, 2]\). This involves integrating the acceleration function over the given interval to find the velocity function, and then finding the difference in velocity at the endpoints of the interval.

Set up the Integral

To find the change in velocity, we integrate the acceleration function \(a(t) = t\) over the interval \([0, 2]\). The integral of the acceleration function gives us the velocity change:\[\Delta v = \int_{0}^{2} a(t) \, dt = \int_{0}^{2} t \, dt.\]

Integrate the Acceleration Function

Find the integral of the function \(t\). The integral of \(t\) is \(\frac{t^2}{2}\). So the definite integral from 0 to 2 is:\[\int_{0}^{2} t \, dt = \left[\frac{t^2}{2}\right]_{0}^{2}.\]

Evaluate the Definite Integral

Substitute the upper limit (\(t=2\)) and lower limit (\(t=0\)) into the integrated function:\[\left.\frac{t^2}{2}\right|_{0}^{2} = \frac{2^2}{2} - \frac{0^2}{2} = \frac{4}{2} - 0 = 2.\]

Interpret the Result

The result of the integral is the change in velocity over the time interval \([0, 2]\). The change in velocity \(\Delta v\) is 2 feet per second.

Key concepts

velocity change

The concept of velocity change is crucial when you want to understand how an object's speed changes over time due to acceleration. In this context, acceleration is a change in velocity per unit time. To find out how much the velocity changes, we look at the integral of the acceleration over a given time interval.

This results in the **definite integral**, which is a value that tells us how much the velocity has changed between two points in time. In this exercise, you are asked to calculate the velocity change of an object moving in a straight line, given its acceleration function (\(a(t) = t\) in \(\text{ft/s}^2\)). By integrating, you find the total change in velocity over the time interval \([0, 2]\).

Simply put, velocity change is the net increase or decrease in an object's speed over time, computed by integrating the acceleration.

integral calculus

Integral calculus is a branch of mathematics focused on accumulation of quantities and the areas under and between curves. In the context of motion, integral calculus allows us to determine the change in velocity when we know the acceleration function.

By taking the integral of an acceleration function, such as \(a(t) = t\), we can find the velocity function. This means we can describe exactly how fast an object will be moving at any moment within a time interval. The essence lies in the summation of infinitely small changes which provides us with the exact change over a larger interval.

This process is reversed compared to differentiation, where you find the rate of change. In integration via integral calculus, you find the whole change based on known rates.

definite integral

A definite integral is used when you want to calculate the overall change between two specified points. In our given example, we sought the change in velocity from time \(t = 0\) to \(t = 2\).

To set up the definite integral, identify the limits of integration (in this case, \(0\) and \(2\)) and the function to integrate (here, \(t\)). The result of this calculation, called the **definite integral**, is a number that represents the total change in the function's value (velocity, in this case) between these limits.

The process includes evaluating the integral and then substituting the limits into the result. You find the net change by subtracting the lower limit outcome from that of the upper limit. Here, this net change was found to be 2 feet per second.

acceleration to velocity

Understanding how acceleration relates to velocity involves moving from knowing how fast an object's speed is changing to knowing its actual speed over time. Simply described, acceleration tells you how the velocity is expected to change.

By using integration, you translate the acceleration function into a velocity function, which reflects the real speed of an object at specific times. For example, with an acceleration function \(a(t) = t\), integrating this function over the interval \([0, 2]\) results in finding the change in velocity over that time period.

The process of going from acceleration to velocity implies understanding the underlying dynamics of motion - how speeding up or slowing down modifies the actual velocity at any given time.