Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

An object travels along a straight line. The function $v(t) = - t^{\frac{1}{5}} + 2t^{-2}$ gives the object's velocity, in feet per minute, at time $t > 0$ minutes. $\newline$ Write a function that gives the object's acceleration $a(t)$ in feet per minute per minute. $\newline$ a(t) = ______

View step-by-step help

View step-by-step help

Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. An object travels along a straight line. The function

v(t) = - t^{\frac{1}{5}} + 2t^{-2}

gives the object's velocity, in feet per minute, at time

t > 0

minutes.

\newline

Write a function that gives the object's acceleration

a(t)

in feet per minute per minute.

\newline

a(t) = ______

Differentiate Power Rule: To find the acceleration function $a(t)$ , we need to differentiate the velocity function $v(t)$ with respect to time $t$ .
Differentiate $2t^{-2}$ : Differentiate each term of $v(t)$ separately. For the first term, $-t^{1/5}$ , use the power rule: $\frac{d}{dt}$ of $t^n = n*t^{n-1}$ . Here, $n = \frac{1}{5}$ .
Combine Derivatives: For the second term, $2t^{-2}$ , again use the power rule.
Find Acceleration Function: Combine the derivatives to get the acceleration function $a(t)$ .

More problems from Relate position, velocity, speed, and acceleration using derivatives

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{0.7} \sin (2 t+3)

. What is the acceleration of the particle at time

t=4

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{1.1} \cos \left(t^{2}-5\right)

. If the position of the particle is

x=-2

t=2.5

, what is the position of the particle when

t=1

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{0.9} \sin (t-1)

. If the position of the particle is

x=-3

t=3

, what is the position of the particle when

t=5

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{2.1} \sin (2 t)

. Find all times when the speed of the particle is equal to

2

on the interval

0 \leq t \leq 4

. You may use a calculator and round your answer to the nearest thousandth.

\newline

t=

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{2.3}+5\right) \sin (2 t)

. What is the distance traveled by the particle over the interval

0 \leq t \leq 5

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{1.9} \cos (3 t-1)

. What is the distance traveled by the particle over the interval

0 \leq t \leq 3

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{1.6}+3\right) \cos (3 t)

. What is the average velocity of the particle on the interval

0 \leq t \leq 6

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its acceleration is given by

a(t)=\left(t^{0.5}+3\right) \cos (3 t)

. If the velocity of the particle is

v=-2

t=1

, what is the velocity of the particle when

t=1

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=t^{2.1} \sin (2 t+5)

. Find all times when the speed of the particle is equal to

1

on the interval

0 \leq t \leq 3

. You may use a calculator and round your answer to the nearest thousandth.

\newline

t=

Posted 2 months ago

Question

A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{2.1}+5\right) \cos (2 t)

. What is the distance traveled by the particle over the interval

0 \leq t \leq 5

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Posted 2 months ago