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# The position of an object with mass $m$ at time $t$ is $\textbf{r}(t) = at^2 \, \textbf{i} + bt^3 \, \textbf{j}$, $0 \leqslant t \leqslant 1$.(a) What is the force acting on the object at time $t$?(b) What is the work done by the force during the time interval $0 \leqslant t \leqslant 1$?

## (a) $2 m a$ i $+6 m b t$ j, $0 \leq t \leq 1$(b) $2 m a^{2}+\frac{9}{2} m b^{2}$

Vector Calculus

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##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

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### Video Transcript

It's a position at Time T is given by this or this force after on the object at time T Well, we know the forces times acceleration which is the second derivative. So sorry. So which is that? They'LL compare the seven the room with you So? So the first derivative is to a t threepeat his square The second derivative is to a six Petey Okay, so So we have this. So this is our force and how this will work down. So if we have thiss on DH, then the work will is integral off If that are, sir Integral will be F d r which is a cemetery in the crate from ah we put in the f the dot product off f in our prime which should be a c i for a squared t plus eighteen p squared He cute t And before I am a square T score over two were probably in one so divided by two eighteen piece where tea fourth over For us, they would probably once So we divide by four So I too am a square plus Ni over to peace. Ah, sorry Hero should have them and b square

#### Topics

Vector Calculus

##### Catherine R.

Missouri State University

##### Heather Z.

Oregon State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp