Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

The rate of change of velocity with respect to time is called acceleration. Suppose that the velocity at time \(t\) of a particle is given by \(v(t)=2 t^{2}\). Find the instantaneous acceleration when \(t=1\) second.

Step-by-Step Solution

Answer

The instantaneous acceleration at \(t=1\) second is 4 m/s².

Step 1: Understand the Concept

The problem asks for an instantaneous acceleration at a specific time. Instantaneous acceleration is the derivative of velocity with respect to time, which represents how fast the velocity of an object is changing at a particular instant.

Step 2: Write Down the Velocity Function

We are given the velocity as a function of time: \[ v(t) = 2t^2 \] This informs us how velocity changes as time changes.

Step 3: Find the Derivative of the Velocity Function

The instantaneous acceleration is the derivative of the velocity function, \(v(t)\), with respect to time \(t\):\[ a(t) = \frac{dv}{dt} = \frac{d}{dt}(2t^2) \]

Step 4: Differentiate Using Power Rule

To differentiate \(v(t) = 2t^2\), apply the power rule, \(\frac{d}{dt}(t^n) = nt^{n-1}\):\[ a(t) = \frac{d}{dt}(2t^2) = 2 \times 2t^{2-1} = 4t \]

Step 5: Evaluate the Derivative at t=1

Substitute \(t=1\) into the derivative to find the instantaneous acceleration:\[ a(1) = 4 \times 1 = 4 \]

Step 6: Conclusion

Thus, the instantaneous acceleration of the particle at \(t=1\) second is determined.

Key Concepts

Understanding Instantaneous Acceleration

Instantaneous acceleration is a key concept in calculus that measures how quickly the velocity of an object is changing at a specific point in time. Think of it as capturing a snapshot of acceleration at an exact moment.
For motion, it's crucial because it helps us understand how an object speeds up or slows down, instantaneously, not just on average over a period.
In mathematical terms, instantaneous acceleration is derived from the velocity function of an object.
It requires calculating the rate at which velocity changes as time progresses — essentially, it’s the derivative of the velocity function with respect to time.
This derivative supplies us with an acceleration function, which, when evaluated at a particular moment,

• Gives a clear answer of how the velocity is changing exactly at that point.
• Provides insights into the motion dynamics, whether an object is speeding up, slowing down, or maintaining speed.

Understanding this helps in predicting future motion and is especially useful in fields like engineering and physics, where knowing the precise motion of parts or particles is essential.

Exploring Derivatives in Calculus

Derivatives are fundamental to calculus and represent the rate at which a quantity changes. In our context, we are concerned with how velocity changes over time.
The derivative of a function gives us an entirely new function, known as the derivative function, that can compute these rates of change at any point in time.
For our velocity function given by:

• \( v(t) = 2t^2 \),
• the derivative can be determined using the power rule.

The power rule states that for a function in the form of \( t^n \), its derivative is \( nt^{n-1} \).
When applying this to the velocity function:

• \( v(t) = 2t^2 \), you derive the acceleration function as:
• \( a(t) = 4t \).

This new function \( a(t) \) can now compute the instantaneous acceleration—showing how rapidly an object's velocity is changing at any single moment.

The Velocity Function and Its Significance

The velocity function in calculus is a representation of an object's speed and direction as it travels over time.
In our example, the velocity function is given by:

• \( v(t) = 2t^2 \), which signifies that velocity changes quadratically as time goes by.

This quadratic relationship implies:

• If time doubles, the velocity quadruples, indicating a rapid increase in speed as time progresses.

By differentiating this function, we can glean further insights into the motion's dynamics.
Velocity functions are essential in physics and engineering because:

• They allow us to study how an object's speed varies, giving us clear insights into its motion path.
• Help in constructing accurate models of real-world scenarios where predicting movement patterns is crucial, such as in vehicle safety technologies.

Understanding the velocity function grants the foundation needed to compute acceleration and further understand the intricacies of motion.