Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Velocity

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #31 : Calculus

Given the following path:

$\left\{\begin{matrix}x(t)=t^3e^{t}\\ y(t)=t^4e^{t} \end{matrix}\right.$

What will be the velocity $\vec{v}$ of someone driving along that path?

Possible Answers:

$((2t^3+t^2)e^{t},(t^4+4t^3)e^{t})$

$((t^3+3t^2)e^{t},(t^4+4t^3)e^{t})$

$((3t^3+3t^2)e^{t},(t^4+4t^3)e^{t})$

$((t^3+3t^2)e^{t},(2t^4+4t^3)e^{t})$

$((t^3+3t^2)e^{t},(3t^4+4t^3)e^{t})$

Correct answer:

$((t^3+3t^2)e^{t},(t^4+4t^3)e^{t})$

Explanation:

To find the velocity we need to differentiate each of the components of the position with respect to time.

We note first that:

$(t^3e^{t})'=(t^3+3t^2)e^{t}$

$(t^4e^{t})'=(t^4+4t^3)e^{t}$ .

The velocity is then given by:

$\vec{v}=((t^3e^{t})',(t^4e^{t})')$ .

Using the above derivatives, we find the velocity:

$\vec{v}=((t^3+3t^2)e^{t},(t^4+4t^3)e^{t})$

Report an Error

Example Question #32 : Calculus

A car is moving following the speed law:

${}\vect{v}=(t^2,t-1,t^3arctan(t))$

What is its acceleration?

Possible Answers:

$\left(2t,1,3t^2arctan(t)+\frac{t^3}{t^3+1}\right)$

$\left(2t,1,3t^2arctan(t^2)+\frac{t^3}{t^2+1}\right)$

No, they will never collide.

$\left(2t,1,3t^2arctan(t)+\frac{t^2}{t^2+1}\right)$

$\left(2t,1,3t^2arctan(t)+\frac{t^3}{t^2+1}\right)$

Correct answer:

$\left(2t,1,3t^2arctan(t)+\frac{t^3}{t^2+1}\right)$

Explanation:

To find the expression for the velocity, we need to differentiate each of the components with respect to time.

We have:

(t^2)'=2t

(t-1)'=1

$(t^3arctan(t))'=3t^2arctan(t)+\frac{t^3}{1+t^2}$

Collecting the components we obtain the expression for the velocity:

$\vec{v}=\left(2t,1,3t^2arctan(t)+\frac{t^3}{t^2+1}\right)$

Report an Error

Example Question #33 : Calculus

The velocity of a particle is given by:

$\vec{V}=\left(\sqrt{3-t},\frac{e^{2t}}{t},ln(t+2)\right)$

When is the above expression defined?

Possible Answers:

(0,3]

(-2,3]

(-2,2)

(-2,3)

(-2,0)

Correct answer:

(0,3]

Explanation:

To have V defined for all t, we need to have each of the components defined as well.

First we see that for the first component to be defined , we need to have $3-t\ge 0$ .

For the second component we need to have $t\ne 0$ .

The last component must satisfy t+2 >0 .

This means that t must satisfy all these 3 restrictions.

This gives :

$\left\{\begin{matrix} t\le 3\\ t>-2 \\t>0 \end{matrix}\right.$

This means that $t\in (0,3]$

Report an Error

Example Question #31 : Velocity

The position of a moving ant is given by:

$\vec{r}=(tsin(t),tcos(t),1)$

Find its speed.

Possible Answers:

t^2

$\sqrt{2+t^2}$

$\sqrt{1-2t^2}}$

$\sqrt{2+3t^2}$

$\sqrt{1+t^2}$

Correct answer:

$\sqrt{1+t^2}$

Explanation:

To find the speed, we need to find the velocity first. We are given the position vector. We need to differentiate to get the velocity vector and find its norm to find the speed.

To find the velocity we differentiate componentwise.

Using the Chain Rule we find the derivative to be:

$\vec{V}=(sin(t)+tcos(t),cos(t)-tsin(t),0)$

We take the norm of the above vector to get :

$||\vec{V}||=\sqrt{sin(t)+tcos(t))^2+(cos(t)-tsin(t))^2}=\sqrt{1+t^2}$

Report an Error

Example Question #32 : Velocity

A car is moving through the following path:

$\vec{r}=(sin(t),sin(t),sin(t),sin(t))$

Find its speed at any given time.

Possible Answers:

2cos(t)

4cos(t)

2sin(t)

2|cos(t))|

4sin(t)

Correct answer:

2|cos(t))|

Explanation:

To find the speed, we need to find the velocity first. We need to differentiate the expression of the position vector componentwise with respect to time.

Knowing that (sin(t))=cos(t) .

We have the velocity given by:

$\vec{V}=(cos(t),cos(t),cos(t),cos(t))$ .

To find the speed we will have to find the norm of $\vec{V}$ .

$||\vec{V}||=\sqrt{cos^2(t)+cos^2(t)+cos^2(t)+cos^2(t)}$

and finally this gives:

$||\vec{V}||=\sqrt{4cos^2(t)}=2|cos(t)|$

Report an Error

Example Question #36 : Calculus

Given the vector position:

$\vec{r}=(2+te^{-t},arctan(t)+1,arctan(2t))$

Find the expression of the velocity.

Possible Answers:

$\left((1-2t)e^{-t},\frac{1}{1+t^2},\frac{2}{4t^2+1}\right)$

$\left((1-t)e^{-t},\frac{1}{1+2t^2},\frac{2}{4t^2+1}\right)$

$\left((1-t)e^{-t},\frac{1}{1+t^2},\frac{2}{4t^2+1}\right)$

$\left((1-t)e^{-t},\frac{1}{1+t^2},\frac{1}{4t^2+1}\right)$

$\left((1-t)e^{-t},\frac{1}{1+t^2},\frac{1}{2t^2+1}\right)$

Correct answer:

$\left((1-t)e^{-t},\frac{1}{1+t^2},\frac{2}{4t^2+1}\right)$

Explanation:

To find the expression of the velocity vector, we must differentiate each of the position vector components with respect to time.

We have then:

$(te^{-t})'=(1-t)e^{-t}$

$(artan(t))'=\frac{1}{1+t^2}$

$(artan(2t))'=\frac{2}{4t^2+1}$

Finally collecting all these derivatives componentwise, we obtain the velocity.

$\left((1-t)e^{-t},\frac{1}{1+t^2},\frac{2}{4t^2+1}\right)$

Report an Error

Example Question #33 : Velocity

Given the position, s(t)=5t^4 , find the velocity at time t=3 .

Possible Answers:

540

6075

243

405

Correct answer:

540

Explanation:

To find the velocity given the position function, take the derivative of the position function using the Power Rule.

$f(x)=x^n \rightarrow f'(x)=nx^{n-1}$

Applying the Power Rule to our function we get the following.

s(t)=5t^4

$v(t)=\frac{ds}{dt}=4\cdot 5 t^{4-1}=20t^3$

Substitute t=3 to determine the velocity at this time.

v(t=3)=20(3)^3=540

Report an Error

Example Question #34 : Velocity

Suppose the position function of an object is s(t)=ln(t) . What is the velocity of the object when $t=\frac{1}{3}$ ?

Possible Answers:

ln(3)

$\frac{1}{3}$

$ln(\frac{1}{3})$

Correct answer:

Explanation:

To find the velocity of s(t) , take the derivative of this function to find s'(t) .

s(t)=ln(t)

Recall that natural log has a special derivative.

$s'(t)=\frac{1}{t}$

Substitute $t=\frac{1}{3}$ and find the velocity at this time.

$s'\left(t=\frac{1}{3}\right)=\frac{1}{\frac{1}{3}}=3$

Report an Error

Example Question #31 : Calculus

The position of an object is described by the function s(t)=3t+2 . What is the average velocity of this object from t=1 to t=6 ?

Possible Answers:

$\frac{17}{5}$

Correct answer:

Explanation:

Write the formula for average velocity.

$v_{avg}= \frac{s_{f}-s_{i}}{t_{f}-t_{i}}$

The variables $s_{f}$ is the final position, $s_{i}$ is the initial position, $t_{f}$ is the final time, and $t_{i}$ is the initial time.

Plug in $t_{i}=1$ and $t_{f}=6$ into s(t)=3t+2 to solve for initial and final position.

$s_{f}= 3(6)+2 =20$

$s_{i}=3(1)+2=5$

Substitute the known variables into the average velocity formula.

$v_{avg}= \frac{20-5}{6-1}=\frac{15}{5}=3$

Report an Error

Example Question #35 : Velocity

Find the velocity function of a particle if the acceleration function is a(t)= -cos(x)+sin(x) .

Possible Answers:

-sin(x)-cos(x)

sin(x)+cos(x)

sin(x)-cos(x)

-sin(x)+cos(x)

sin(x)+cos(x)

Correct answer:

-sin(x)-cos(x)

Explanation:

To obtain the velocity function, integrate the acceleration function a(t)= -cos(x)+sin(x) .

$v(t)=\int(-cos(x)+sin(x))dx=-sin(x)-cos(x)$

Report an Error

View Calculus Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

View Calculus Tutors

Travis
Certified Tutor

Delaware State University, Bachelor of Science, Mathematics.

View Calculus Tutors

Shiblu
Certified Tutor

University of Central Florida, Doctor of Science, Civil Engineering.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Math Tutors in Miami, Statistics Tutors in San Diego, GRE Tutors in Washington DC, Physics Tutors in New York City, Spanish Tutors in Boston, Chemistry Tutors in Los Angeles, Physics Tutors in Denver, SSAT Tutors in Boston, ACT Tutors in Houston, Statistics Tutors in Dallas Fort Worth

Popular Courses & Classes

LSAT Courses & Classes in Houston, ACT Courses & Classes in Phoenix, ACT Courses & Classes in San Diego, LSAT Courses & Classes in Chicago, LSAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Denver, MCAT Courses & Classes in Chicago, ACT Courses & Classes in Houston, SAT Courses & Classes in Chicago, LSAT Courses & Classes in San Francisco-Bay Area

Popular Test Prep

LSAT Test Prep in Los Angeles, GMAT Test Prep in Denver, GRE Test Prep in Los Angeles, ISEE Test Prep in Philadelphia, ACT Test Prep in San Diego, LSAT Test Prep in Dallas Fort Worth, ACT Test Prep in Denver, SSAT Test Prep in Boston, GMAT Test Prep in Los Angeles, MCAT Test Prep in San Diego

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 1 : Velocity

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #31 : Calculus

Example Question #32 : Calculus

Example Question #33 : Calculus

Example Question #31 : Velocity

Example Question #32 : Velocity

Example Question #36 : Calculus

Example Question #33 : Velocity

Example Question #34 : Velocity

Example Question #31 : Calculus

Example Question #35 : Velocity

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: