Problem 161 - Calculus Volume 3

Question

Find the velocity, acceleration, and speed of a particle with the given position function. \(\mathbf{r}(t)=\left\langle e^{t}, e^{-t}\right\rangle\)

Step-by-Step Solution

Answer

Velocity: \( \langle e^{t}, -e^{-t} \rangle \), Acceleration: \( \langle e^{t}, e^{-t} \rangle \), Speed: \( \sqrt{e^{2t} + e^{-2t}} \).

Step 1: Find the Velocity Function

The velocity function is the derivative of the position function with respect to time. The position function is given as \( \mathbf{r}(t) = \left\langle e^{t}, e^{-t}\right\rangle \). Therefore, to find the velocity, differentiate each component separately:\[ \mathbf{v}(t) = \left\langle \frac{d}{dt}(e^{t}), \frac{d}{dt}(e^{-t}) \right\rangle = \left\langle e^{t}, -e^{-t} \right\rangle \]

Step 2: Find the Acceleration Function

The acceleration function is the derivative of the velocity function with respect to time. We found the velocity function to be \( \mathbf{v}(t) = \left\langle e^{t}, -e^{-t} \right\rangle \). Differentiate each component of the velocity function:\[ \mathbf{a}(t) = \left\langle \frac{d}{dt}(e^{t}), \frac{d}{dt}(-e^{-t}) \right\rangle = \left\langle e^{t}, e^{-t} \right\rangle \]

Step 3: Find the Speed at Time t

Speed is the magnitude of the velocity vector. Using the velocity function \( \mathbf{v}(t) = \left\langle e^{t}, -e^{-t} \right\rangle \), calculate the magnitude:\[ \text{Speed} = \sqrt{(e^{t})^2 + (-e^{-t})^2} = \sqrt{e^{2t} + e^{-2t}} \]

Step 4: Conclusion

We found the velocity, acceleration, and speed for the given particle's position function. The velocity is \( \left\langle e^{t}, -e^{-t} \right\rangle \), acceleration is \( \left\langle e^{t}, e^{-t} \right\rangle \), and speed is \( \sqrt{e^{2t} + e^{-2t}} \).

Key Concepts

Velocity Function

In vector calculus, the velocity function represents how the position of a particle changes over time. For a position function like \( \mathbf{r}(t) = \langle e^{t}, e^{-t} \rangle \), the velocity function is the derivative of \( \mathbf{r}(t) \) with respect to time \( t \). This means we take the derivative of each component of the vector separately.
To find the velocity function \( \mathbf{v}(t) \), we compute:

• The derivative of \( e^{t} \) is \( e^{t} \).
• The derivative of \( e^{-t} \) is \( -e^{-t} \).

Thus, the velocity function is \( \mathbf{v}(t) = \langle e^{t}, -e^{-t} \rangle \). This vector shows the rate of change of each component of the position as the particle moves.
This function is crucial because it helps us understand the direction and speed at which a particle's position changes over time.

Acceleration Function

The acceleration function in vector calculus describes how the velocity of a particle changes with time. It is essentially the derivative of the velocity function. For our velocity function, \( \mathbf{v}(t) = \langle e^{t}, -e^{-t} \rangle \), the acceleration function is obtained by differentiating each component with respect to \( t \).
Let's compute:

• The derivative of \( e^{t} \) is again \( e^{t} \).
• The derivative of \( -e^{-t} \) is \( e^{-t} \), considering the chain rule with \( -1 \).

Thus, the acceleration function is \( \mathbf{a}(t) = \langle e^{t}, e^{-t} \rangle \).
Acceleration is a key concept as it not only tells us whether the particle is speeding up or slowing down but also provides insight into changes in the trajectory's direction.

Magnitude of Velocity

The magnitude of velocity, often referred to as speed, is the length of the velocity vector. It tells us how fast the particle is moving regardless of direction. For the velocity function \( \mathbf{v}(t) = \langle e^{t}, -e^{-t} \rangle \), the speed is calculated by finding the magnitude of this vector.
To compute the speed:

• Square each component: \((e^{t})^2 + (-e^{-t})^2\).
• Sum the squares: \(e^{2t} + e^{-2t}\).
• Take the square root of the sum: \(\sqrt{e^{2t} + e^{-2t}}\).

This gives us the speed of the particle at time \( t \).
Understanding the magnitude of velocity is important when you want to know just the rate at which the particle is moving, without considering the specific direction of its motion.