- Answer the following:
- Find the maximum speed of the object and the time it takes.
- What is the minimum speed of the object along with the time it takes?
- t is the time interval $[0,4]$ in seconds.
This problem aims to find the maximum speed of an object that covers a distance in the shape of a parameterized curve whose equations are given.
To better understand the problem, you must be familiar with the parameterized curve in a plane, terminal, and initial velocities. A parametrized curve is a trail in the $xy$ plane outlined by the point $x(t), y(t)$ as the parameter $t$ spans over an interval $I$.
The set builder notation for curve will be:
\[c = \{ (x(t), y(t)) \colon t \in I \}\]
Expert Answer
We are given the following two equations of the object that is moving along a parametrized curve:
\[x(t) = e^t + e^{-t} \]
\[ y(t) = e^{-t} \]
$[0, 4]$ is the time interval $t$.
Position vector at time $t$ will be:
\[ R(t) = <x(t),y(t)> = <e^t + e^{-t}, e^{-t} >\]
Velocity vector at time $t$ is:
\[ v(t) = \dfrac{d}{dt} R(t) \]
\[ = \dfrac{d}{d_t} < e^t + e^{-t}, e^{-t} > \]
\[ v(t) = < e^t – e^{-t}, – e^{-t} > \]
Scalar speed at time $t$ comes out to be:
\[ v(t) = |v(t)| = |< e^t – e^{-t}, – e^{-t} >| \]
\[ = \sqrt{(e^t – e^{-t})^2 + e^{-2t}} \]
\[ = \sqrt{e^{2t} + e^{2t} -2 + e^{-2t}} \]
\[ v(t) = \sqrt{e^{2t} + 2e^{-2t} -2 } \]
Consider the function,
\[ f(t) = \sqrt{e^{2t} + 2e^{-2t} -2 } \]
\[ f'(t) = \dfrac{e^{2t}-2e^{-2t}} {\sqrt{e^{2t} + 2e^{-2t} -2 }} \]
For minima or maxima,
\[ f'(t) = 0 \]
\[ \dfrac{e^{2t}-2e^{-2t}} {\sqrt{e^{2t} + 2e^{-2t} -2 }} = 0 \]
\[ e^{2t}-2e^{-2t} = 0 \]
\[ e^{4t} = 2 \]
\[ 4t = ln(2) \]
\[ t = \dfrac{1}{4}ln(2) \]
$\dfrac{1}{4}ln(2)$ is the critical point of $f$.
End points and critical points are found as follows:
\[ f(t) = \sqrt{e^{2t} + 2e^{-2t} -2 } \]
\[ f(0) = \sqrt{e^{2(0)} + 2e^{-2(0)} -2 } = 1 \]
\[ f(4) = \sqrt{e^{2(4)} + 2e^{-2(4)} -2 } = 54.58 \]
\[ f(\dfrac{1}{4}ln(2)) = \sqrt{\sqrt{2} + 2 \left( \dfrac{\sqrt{2}}{2} \right) -2 } \]
\[ = \sqrt{2\sqrt{2} -2 } = 0.91 \]
Thus, the Maximum speed at interval $4$ is $54.58$,
Whereas the Minimum speed at interval $f(\dfrac{1}{4}ln(2))$ is $0.91$.
Numerical Result
The maximum speed of the object on the time interval is $54.58$ at time $t=4$.
The minimum speed of the object on the time interval is $0.91$ at time $t=f(\dfrac{1}{4}ln(2))$.
Example
We are given the following two equations of the object that is moving along a parametrized curve:
\[x(t) = e^t + e^{-t}\]
\[y(t) = e^{-t}\]
Finding the speed on the interval $t=2$:
\[f(t) = \sqrt{e^{2t} + 2e^{-2t} -2 } \]
\[f(2) = \sqrt{e^{2(2)} + 2e^{-2(2)} -2 } = 7.25 \]
The speed of the object on the time interval is $7.25$ at time $t=2$.