Question

Find the displacement, distance traveled, average velocity and average speed of the described object on the given interval. An object with velocity function \(\vec{v}(t)=\langle\cos t, \sin t\rangle,\) where distances are measured in feet and time is in seconds, on \([0,2 \pi]\)

Short answer

Displacement: \(\langle 0, 0 \rangle\), Distance traveled: \(2\pi\) feet, Average velocity: \(\langle 0, 0 \rangle\), Average speed: \(1\) foot/sec.

Step-by-step solution

Calculate Displacement

The displacement is the net change in position, which can be determined by integrating the velocity vector over the time interval.\[ \text{Displacement} = \int_{0}^{2\pi} \vec{v}(t) \ dt = \int_{0}^{2\pi} \langle \cos t, \sin t \rangle \ dt \]Integrating each component separately:For \(\cos t\):\[ \int_{0}^{2\pi} \cos t \ dt = \sin t \Big|_{0}^{2\pi} = \sin(2\pi) - \sin(0) = 0 \]For \(\sin t\):\[ \int_{0}^{2\pi} \sin t \ dt = -\cos t \Big|_{0}^{2\pi} = -\cos(2\pi) - (-\cos(0)) = 1 - 1 = 0 \]Thus, the displacement is \( \langle 0, 0 \rangle \).

Calculate Distance Traveled

The distance traveled is equal to the integral of the magnitude of the velocity vector over the interval.First, find the magnitude of the velocity vector:\[ \| \vec{v}(t) \| = \sqrt{(\cos t)^2 + (\sin t)^2} = \sqrt{1} = 1 \]Now, integrate the constant magnitude over \([0, 2\pi ]\):\[ \text{Distance Traveled} = \int_{0}^{2\pi} 1 \, dt = t \Big|_{0}^{2\pi} = 2\pi - 0 = 2\pi \]Thus, the distance traveled is \(2\pi\) feet.

Determine Average Velocity

Average velocity is the total displacement divided by the total time.\[ \text{Average Velocity} = \frac{\text{Displacement}}{\text{Time Interval}} = \frac{\langle 0, 0 \rangle}{2\pi - 0} = \langle 0, 0 \rangle \]Therefore, the average velocity is \( \langle 0, 0 \rangle \) feet per second.

Calculate Average Speed

Average speed is the total distance traveled divided by the total time.\[ \text{Average Speed} = \frac{\text{Distance Traveled}}{\text{Time Interval}} = \frac{2\pi}{2\pi} = 1 \]So, the average speed is \(1\) foot per second.

Key concepts

Displacement

Displacement is a fundamental concept in calculus and physics that describes the change in position of an object. It is defined as the net change from the starting point to the ending point of an object's motion. In essence, displacement takes into account both the path traveled and the direction. To find displacement when dealing with a velocity function, we integrate the velocity vector over the given interval. For example, with a velocity function \[ \vec{v}(t) = \langle \cos t, \sin t \rangle \]on the interval \([0, 2\pi]\),the displacement is calculated by \[ \int_{0}^{2\pi} \vec{v}(t)\, dt = \int_{0}^{2\pi} \langle \cos t, \sin t \rangle\, dt \]. Integrating each component separately:- For \(\cos t\), the integral result is zero because \(\sin(2\pi) = \sin(0) = 0\).- For \(\sin t\), it results in zero because \(-\cos t\) evaluated from \(0\) to \(2\pi\) gives 0.Thus, the displacement is \(\langle 0, 0 \rangle\). This means the object has returned to its starting position.

Distance Traveled

Distance traveled describes the total length of the path taken by an object, regardless of its starting or ending point. Unlike displacement, distance does not account for direction. It simply measures how much ground has been covered.To find the distance traveled from a velocity function, you need to integrate the magnitude of the velocity vector over the given time interval. For velocity function \(\vec{v}(t) = \langle \cos t, \sin t \rangle\),first calculate the vector's magnitude:\[\| \vec{v}(t) \| = \sqrt{(\cos t)^2 + (\sin t)^2} = \sqrt{1} = 1\]Since the magnitude is constantly \(1\), integrate this over \([0, 2\pi]\) to determine the total distance:\[\text{Distance Traveled} = \int_{0}^{2\pi} 1 \, dt = t \Big|_{0}^{2\pi} = 2\pi\]Thus, the total distance traveled is \(2\pi\) feet, indicating the object has completed a full loop and returned to its starting position.

Average Velocity

Average velocity offers insight into the overall change in position over a period of time. It is the ratio of total displacement to total time. Even if the object moves in different directions, average velocity only considers the start and end points.To calculate average velocity:- Determine the displacement, which in this case is\(\langle 0, 0 \rangle\),as the object returns to its original position.- Calculate the total time from the interval which is \(2\pi\)(with seconds as the unit).Therefore, \[\text{Average Velocity} = \frac{\langle 0, 0 \rangle}{2\pi} = \langle 0, 0 \rangle\]This indicates that the average velocity is \(\langle 0, 0 \rangle\)feet per second, signifying no net movement in any particular direction over the interval.

Average Speed

Average speed measures how quickly an object is moving, averaged over the time interval. Unlike average velocity, it considers the total distance traveled rather than displacement, making it always non-negative.To find the average speed:- Use the total distance traveled, which is \(2\pi\)feet,calculated previously.- Divide this by the total time, \(2\pi\)seconds, for the interval:\[\text{Average Speed} = \frac{2\pi}{2\pi} = 1\]The average speed is \(1\)foot per second. This represents a constant speed where the object circulates and completes a cycle, regardless of its back-and-forth movement.