Problem 62
Question
Assume that a particle's position on the \(x\) -axis is given by $$x=3 \cos t+4 \sin t,$$ where \(x\) is measured in feet and \(t\) is measured in seconds. a. Find the particle's position when \(t=0, t=\pi / 2,\) and \(t=\pi\). b. Find the particle's velocity when \(t=0, t=\pi / 2,\) and \(t=\pi\).
Step-by-Step Solution
Verified Answer
Position: 3 feet at \(t=0\), 4 feet at \(t=\frac{\pi}{2}\), -3 feet at \(t=\pi\). Velocity: 4 ft/s at \(t=0\), -3 ft/s at \(t=\frac{\pi}{2}\), -4 ft/s at \(t=\pi\).
1Step 1: Evaluate Position Function at t=0
Substitute \( t = 0 \) into the position function \( x = 3 \cos t + 4 \sin t \). We get: \[ x = 3 \cos(0) + 4 \sin(0) = 3 \times 1 + 4 \times 0 = 3 \text{ feet} \]
2Step 2: Evaluate Position Function at t=π/2
Substitute \( t = \frac{\pi}{2} \) into the position function \( x = 3 \cos t + 4 \sin t \). We have: \[ x = 3 \cos\left(\frac{\pi}{2}\right) + 4 \sin\left(\frac{\pi}{2}\right) = 3 \times 0 + 4 \times 1 = 4 \text{ feet} \]
3Step 3: Evaluate Position Function at t=π
Substitute \( t = \pi \) into the position function \( x = 3 \cos t + 4 \sin t \). We find: \[ x = 3 \cos(\pi) + 4 \sin(\pi) = 3 \times (-1) + 4 \times 0 = -3 \text{ feet} \]
4Step 4: Differentiate Position Function for Velocity
The velocity function \( v(t) \) is the derivative of the position function \( x(t) \). Differentiate: \[ v(t) = \frac{d}{dt}[3 \cos t + 4 \sin t] = -3 \sin t + 4 \cos t \]
5Step 5: Evaluate Velocity Function at t=0
Substitute \( t = 0 \) into the velocity function \( v(t) = -3 \sin t + 4 \cos t \). We get: \[ v = -3 \sin(0) + 4 \cos(0) = -3 \times 0 + 4 \times 1 = 4 \text{ feet/second} \]
6Step 6: Evaluate Velocity Function at t=π/2
Substitute \( t = \frac{\pi}{2} \) into the velocity function. We have: \[ v = -3 \sin\left(\frac{\pi}{2}\right) + 4 \cos\left(\frac{\pi}{2}\right) = -3 \times 1 + 4 \times 0 = -3 \text{ feet/second} \]
7Step 7: Evaluate Velocity Function at t=π
Substitute \( t = \pi \) into the velocity function: \[ v = -3 \sin(\pi) + 4 \cos(\pi) = -3 \times 0 + 4 \times (-1) = -4 \text{ feet/second} \]
Key Concepts
Particle MotionPosition FunctionVelocity Function
Particle Motion
Particle motion, as it pertains to calculus, refers to the movement of a particle along a certain path, which can be described functionally in terms of time. Understanding particle motion involves determining how the particle's position changes over time, and how fast it moves, known as its velocity.
One key aspect of analyzing particle motion is using functions to represent the position of the particle at different times. By substituting different values of time into these functions, you can determine the exact position of the particle at those moments.
In our exercise, the position function shows how a particle moves along the x-axis. To fully understand the particle's journey, you'd substitute specific times into the position function, producing various positions the particle occupies during its motion.
One key aspect of analyzing particle motion is using functions to represent the position of the particle at different times. By substituting different values of time into these functions, you can determine the exact position of the particle at those moments.
In our exercise, the position function shows how a particle moves along the x-axis. To fully understand the particle's journey, you'd substitute specific times into the position function, producing various positions the particle occupies during its motion.
Position Function
A position function describes the location of a particle at any given time. In calculus, evaluating this function at specific times helps determine the particle's position at those moments.
In our exercise, the position function is given by the formula: \[ x = 3 \cos t + 4 \sin t \]This equation tells us where the particle is located on the x-axis as time rolls on.
The Steps to Evaluate Position:
This concept not only tells us how far the particle has traveled but also forms the basis for finding other properties like velocity.
In our exercise, the position function is given by the formula: \[ x = 3 \cos t + 4 \sin t \]This equation tells us where the particle is located on the x-axis as time rolls on.
The Steps to Evaluate Position:
- Substitute the given time values into the position function.
- Compute the trigonometric values using known identities. For example, we know \( \cos(0) = 1 \) and \( \sin(0) = 0 \).
- Solve the arithmetic to find the particle's position at the specified times.
This concept not only tells us how far the particle has traveled but also forms the basis for finding other properties like velocity.
Velocity Function
The velocity of a particle is the rate at which its position changes over time. To find the velocity function from a position function, you differentiate the position function with respect to time.
Derivative Connections:
Evaluating Velocity:
Derivative Connections:
- The derivative of \( \cos(t) \) is \( -\sin(t) \).
- The derivative of \( \sin(t) \) is \( \cos(t) \).
- These lead to the velocity function \( v(t) = -3 \sin t + 4 \cos t \).
Evaluating Velocity:
- At \( t=0 \), the particle's velocity is \( 4 \) feet/second using \( v = -3 \sin(0) + 4 \cos(0) \).
- At \( t=\pi / 2 \), the velocity reaches \( -3 \) feet/second.
- Finally, at \( t = \pi \), the velocity evaluates to \( -4 \) feet/second.
Other exercises in this chapter
Problem 62
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