Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

The velocity of an object is given by \(3 \sin (\pi t)\). (a) What is the object's speed as a function of time? (b) What is the object's net displacement from \(t=0\) to \(t=2 ?\) (c) How far has the object traveled from \(t=0\) to \(t=2 ?\) (d) What is the object's average velocity on \([0,2] ?\) (e) What is the object's average speed on \([0,2]\) ?

Step-by-Step Solution

Answer

The object's speed function is \(3 \sin (\pi t)\). The net displacement from \(t=0\) to \(t=2\) is 0. The total distance traveled from \(t=0\) to \(t=2\) is 6. The average velocity over the interval \([0,2]\) is 0. The average speed over the interval \([0,2]\) is 3.

Step 1: Determine the speed function

The speed of an object is the absolute value of its velocity, so the speed function is given by \(|3 \sin (\pi t)|\). As the sine function oscillates between -1 and 1, the absolute value of the speed will just be the same function without the sine changing signs, so \(3 \sin (\pi t)\).

Step 2: Compute the object's net displacement

The displacement of an object is calculated by integrating the velocity function. Therefore, to find the net displacement from \(t=0\) to \(t=2\), we need to compute the integral of the velocity function from 0 to 2: \(\int_{0}^{2} 3 \sin (\pi t) dt\). Evaluating this gives 0. Thus, the net displacement is 0.

Step 3: Calculate the total distance travelled

The total distance traveled by an object is found by integrating the speed function. So, to find the total distance travelled from \(t=0\) to \(t=2\), we calculate \(\int_{0}^{2} |3 \sin(\pi t)| dt = \int_{0}^{2} 3 \sin(\pi t) dt\). After performing the integral, we get a total travelled distance of 6.

Step 4: Compute the object's average velocity

The average velocity is given by the total displacement divided by total time. As the total displacement from \(t=0\) to \(t=2\) is 0, the average velocity over the interval \([0,2]\) is therefore \(0/2=0\).

Step 5: Calculate the object's average speed

The average speed is given by the total distance travelled divided by the total time. Therefore, the average speed over the interval \([0,2]\) is \(6/2=3\).

Key Concepts

Speed as a Function of Time

When we talk about speed as a function of time, we're exploring how an object's speed changes over time. In calculus, this means we are looking at a mathematical function that describes speed in relation to time. For our exercise, the velocity of an object is given by the equation \(3 \sin (\pi t)\), and speed is simply the magnitude of velocity—indicating how fast the object is moving without considering its direction.

For positive and negative values of the sine function, speed remains a positive quantity. Hence, the speed function, being the absolute value of the velocity, is expressed as \(|3 \sin (\pi t)|\). This function will always yield a non-negative value for any time \(t\), showing us how the object's speed varies with time, regardless of the direction of motion.

Net Displacement

The concept of net displacement refers to the overall change in position of an object, considering the direction of movement. It is essentially the difference between an object's final and initial position. In terms of calculus, this is obtained by integrating the velocity function over a given time interval.

For the given function, the integration of the velocity \(3 \sin (\pi t)\) from \(t=0\) to \(t=2\) equals zero. This means that the object has returned to its starting position after two units of time, resulting in a net displacement of zero. Net displacement can be zero even if the object has traveled a certain distance, provided it returns back to the starting point.

Total Distance Traveled

The total distance traveled by an object, unlike net displacement, does not consider the direction of travel. It is the sum of all distances the object has covered, and in calculus, this is determined by integrating the speed function. Since speed takes only positive values, the distance keeps accumulating.

In the case of our object, the total distance traveled from \(t=0\) to \(t=2\) is found by integrating \(|3 \sin(\pi t)|\), which yields the result of 6. This figure represents the path length along the trajectory of the object, regardless of its changes in direction during the motion.

Average Velocity

Average velocity is defined as the net displacement divided by the time taken. It's a vector quantity, which means it has both magnitude and direction.

For the given time period from \(t=0\) to \(t=2\), the average velocity can be calculated as the ratio of the net displacement (which we found to be zero) to the total time span (which is 2). This straightforward computation gives us an average velocity value of \(0/2 = 0\), which implies that on average, the position of the object did not change over the interval \([0,2]\), despite it having moved in the meantime.

Average Speed

While average speed may sound similar to average velocity, it is quite different. Average speed is a scalar quantity representing the total distance traveled divided by the time taken, and doesn't involve direction.

In our example, the total distance traveled over the interval \([0,2]\) is 6 units, and the time elapsed is 2 units, rendering an average speed of \(6/2 = 3\) units of speed. This value gives us an idea of how fast the object moved on average over the entire time interval, disregarding any changes in direction.