Question

Find the displacement, distance traveled, average velocity and average speed of the described object on the given interval. An object with position function \(\vec{r}(t)=\langle 5 \cos t,-5 \sin t\rangle\), where distances are measured in feet and time is in seconds, on \([0, \pi]\).

Short answer

Displacement: \(\langle -10, 0 \rangle\); Distance: \(5\pi\) feet; Avg. velocity: \(\langle -\frac{10}{\pi}, 0 \rangle\); Avg. speed: 5 ft/s.

Step-by-step solution

Calculate Displacement

The displacement of the object is the change in its position vector from the start to the end of the time interval. It is calculated as \( \vec{r}(t_2) - \vec{r}(t_1) \), where \( t_1 = 0 \) and \( t_2 = \pi \).\- \( \vec{r}(0) = \langle 5 \cos(0), -5 \sin(0) \rangle = \langle 5, 0 \rangle \)\- \( \vec{r}(\pi) = \langle 5 \cos(\pi), -5 \sin(\pi) \rangle = \langle -5, 0 \rangle \)\The displacement is the difference between these vectors: \( \langle -5, 0 \rangle - \langle 5, 0 \rangle = \langle -10, 0 \rangle \).

Calculate Distance Traveled

The distance traveled is the length of the path taken by the object, which is a semicircle in this case since the position function describes a circular motion. The radius of the circle is 5 feet. Hence, the distance traveled is half the circumference of the circle.\- Circumference of full circle = \( 2\pi\times5 = 10\pi \) feet\- Distance traveled (semicircle) = \( \frac{10\pi}{2} = 5\pi \) feet.

Calculate Average Velocity

The average velocity is the displacement divided by the time interval. Using the displacement and interval from Step 1: \( \langle -10, 0 \rangle \) over the interval \([0, \pi]\).\- \( \text{Displacement} = \langle -10, 0 \rangle \)\- \( \text{Time interval} = \pi - 0 = \pi \)\Average velocity = \( \frac{\langle -10, 0 \rangle}{\pi} = \langle -\frac{10}{\pi}, 0 \rangle \).

Calculate Average Speed

The average speed is the total distance traveled divided by the time it takes. From Step 2, the distance traveled is \(5\pi\) feet, and the time interval is \(\pi\) seconds.\- Average speed = \(\frac{5\pi}{\pi} = 5\) feet per second.

Key concepts

Displacement

In vector calculus, displacement is a vector quantity that describes the change in position of an object. It considers the straight line from the initial position to the final position, without taking into account the path actually traveled. This concept highlights the difference between displacement and the path length, which can be more curved and lengthy.

• Displacement is represented as the difference between the vectors at two points in time.
• For the given exercise, the initial position vector at time zero is \( \vec{r}(0) = \langle 5, 0 \rangle \).
• The final position vector at time \( \pi \) seconds is \( \vec{r}(\pi) = \langle -5, 0 \rangle \).
• By subtracting these, the displacement becomes \( \langle -10, 0 \rangle \) feet.

Distance Traveled

Distance traveled differs from displacement because it accounts for the total length of the path literally taken by an object. In our exercise, the object moves in a circular arc which constitutes half the full circle. The calculation uses the circumference of the complete circle to determine the arc’s length.

• The circle's radius is 5 feet, so the full circumference is \( 2\pi \times 5 = 10\pi \).
• The object moves along half the circle's circumference, as it describes a semicircle.
• Thus, the distance traveled is half that perimeter, resulting in a path length of \( 5\pi \) feet.

Average Velocity

Average velocity provides insight into the velocity over the whole interval, considering only the initial and final position. Unlike average speed, velocity is a vector, encompassing both a magnitude and direction. Thus, it describes how fast an object is moving toward the end destination.

• Average velocity is calculated using the displacement from the exercise: \( \langle -10, 0 \rangle \).
• This vector is divided by the total time interval, here \( \pi \) seconds.
• Consequently, the average velocity becomes \( \langle -\frac{10}{\pi}, 0 \rangle \), indicating it moves in the negative x-direction at that computed rate.

Average Speed

Average speed measures how fast an object is moving, without regard to the direction of motion. It is a scalar quantity, meaning it has only magnitude and no direction. This differs from velocity as speed does not account for displacement but instead considers the entire path's length.

• In this case, the distance traveled was found to be \( 5\pi \) feet (half the circle's complete route).
• The total time taken was \( \pi \) seconds.
• So, the average speed, the total distance captured over time, results in \( \frac{5\pi}{\pi} = 5 \) feet per second.