Question

We will see in a later chapter that the acceleration of a point is the rate of change of the velocity of the point. If the velocity of the arm of an industrial robot is given by \(v=3.45\left(t^{2}+2\right)^{2} \mathrm{ft} / \mathrm{s},\) where \(t\) is the time in seconds, take the derivative of this velocity to find the acceleration, and evaluate it at \(t=1.00 \mathrm{s}.\)

Short answer

The acceleration of the robot arm at \(t = 1.00 \mathrm{s}\) is \(27.60 \mathrm{ft}/\mathrm{s}^2\).

Step-by-step solution

Differentiate the velocity function with respect to time

To find the acceleration, which is the rate of change of velocity, take the derivative of the velocity function, using the chain rule for differentiation. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function, keeping the inner function the same, times the derivative of the inner function. The velocity function is given by \(v = 3.45(t^2 + 2)^2\), so applying the chain rule, let \(u = t^2 + 2\), so the function becomes \(v = 3.45u^2\). Differentiate this with respect to \(t\) by first differentiating with respect to \(u\) and then multiplying it by the derivative of \(u\) with respect to \(t\).

Evaluate the derivative at \(t = 1.00 s\)

After finding the expression for the acceleration, plug in the value of \(t = 1.00 s\) to find the specific acceleration at that time. The derivative from the previous step will provide a formula for acceleration as a function of time that needs to be evaluated at the given point.

Simplify to find the numerical value of the acceleration

Once the expression of the derivative is evaluated at \(t = 1.00 s\), simplify the result to get the acceleration at that specific time in numerical form. This is the acceleration of the robot arm at one second.

Key concepts

Rate of Change

The concept of rate of change is fundamental in calculus and it represents how a quantity, such as velocity, changes over time. In our robot arm scenario, the velocity of the arm changes as time progresses, and understanding this change is key to optimizing its function.

Rate of change can be visualized as the slope of the tangent line on a graph at any given point. For example, if we plot the velocity of the robot arm against time, the slope of the line at any particular point will represent the robot arm's acceleration at that moment. Acceleration is simply the rate of change of velocity.

Thus, when we seek to calculate acceleration at a particular time (like at t=1.00 s for the robot arm), we are finding how steeply the velocity is increasing or decreasing at that point, which is the essence of rate of change in calculus.

Chain Rule Differentiation

The chain rule is a powerful tool in calculus used to differentiate composite functions. A composite function is a function that can be decomposed into two or more functions, where one function is applied to the result of another.

In our example, the function for the velocity of the robot arm, given by \(v = 3.45(t^2 + 2)^2\), is a composite function because we can consider \(u = t^2 + 2\) as one function inside another function \(v = 3.45u^2\). To apply the chain rule, differentiate the outer function with respect to \(u\), which yields \(2 \times 3.45u\). Next, multiply that by the derivative of \(u\) with respect to \(t\), or \(2t\).

The product of these derivatives offers the acceleration formula, smoothly connecting the rate of change of velocity with respect to time. By breaking down complex functions into simpler components, the chain rule methodically finds the derivative that is crucial for calculating the acceleration of the robot arm.

Velocity Function Calculus

In calculus, velocity functions represent how the position of an object changes over time, which is the first derivative of the position function with respect to time. To paint a better picture, consider a car moving along a road; the velocity function would depict the car's speed at any given moment. That speed is the first indication of the car's dynamic state.

For the industrial robot arm described in our exercise, the velocity function is more complex due to the squared term. Nonetheless, the principles remain the same. The velocity function enables us to calculate not just how fast the robot arm is moving but, through differentiation, how its speed changes—in other words, its acceleration.

Through differentiation of this function, we obtain an expression for acceleration. To achieve this, we apply rules of calculus, including the aforementioned chain rule, to manipulate the function into a form where the rate of change at any time \(t\) becomes clear. This transition from the velocity function to the acceleration function is an elegant demonstration of calculus in motion, revealing how dynamic systems like a robot arm can be mathematically analyzed and understood.