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 $v(t) = 2\sqrt{t} - 6\sqrt[3]{t}$ gives the object's velocity, in meters per second, at time $t$ seconds. $\newline$ Write a function that gives the object's acceleration $a(t)$ in meters per second per second. $\newline$ 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) = 2\sqrt{t} - 6\sqrt[3]{t}

gives the object's velocity, in meters per second, at time

t

seconds.

\newline

Write a function that gives the object's acceleration

a(t)

in meters per second per second.

\newline

a(t) = ______

Identify velocity function: Identify the velocity function and recognize the need to differentiate it to find acceleration. $\newline$ $v(t) = 2\sqrt{t} - 6\sqrt[3]{t}$ $\newline$ We need to differentiate $v(t)$ to find $a(t)$ , the acceleration function.
Differentiate each term: Differentiate each term of $v(t)$ separately. $\newline$ For the first term, $2\sqrt{t}$ , the derivative is $2 \times (1/2) \times t^{-1/2} = t^{-1/2}$ . $\newline$ For the second term, $-6\sqrt[3]{t}$ , the derivative is $-6\sqrt[3]{},$ since the derivative of $t$ with respect to $t$ is $1$ .

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