Physics with Calculus
When finding a position vector from a velocity vector using an indefinite integral, what must be included?
A rover moves with velocity V→(t)=(3t2−2)i^+(4t+1)j^\(\overrightarrow{V}\)\(\left\)(t\(\right\))=\(\left\)(3t^2-2\(\right\))\(\hat{i}\)+\(\left\)(4t+1\(\right\))\(\hat{j}\). What is its displacement from t=0t = 0 to t=2t = 2?
A submersible follows velocity V→(t)=(3t2−6t)i^+(4t+2)j^\(\overrightarrow{V}\)\(\left\)(t\(\right\))=\(\left\)(3t^2-6t\(\right\))\(\hat{i}\)+\(\left\)(4t+2\(\right\))\(\hat{j}\). If R→(2)=5i^+j^\(\overrightarrow{R}\)\(\left\)(2\(\right\))=5\(\hat{i}\)+\(\hat{j}\), what is its position vector R→(t)\(\overrightarrow{R}\)\(\left\)(t\(\right\))?
When differentiating a vector function such as r→(t)=x(t)i^+y(t)j^\(\overrightarrow{r}\)\(\left\)(t)\(\right\).=x\(\left\)(t\(\right\))\(\hat{i}\)+y\(\left\)(t\(\right\))\(\hat{j}\), what role do the unit vectors i^\(\hat{i}\) and j^\(\hat{j}\) play?
A delivery robot experiences acceleration a→(t)=4i^+(2t)j^\(\overrightarrow{a}\)\(\left\)(t\(\right\))=4\(\hat{i}\)+\(\left\)(2t\(\right\))\(\hat{j}\) from t=0t = 0 to t=3 st=3\(\text{ s}\). What is the magnitude of its change in velocity over that interval?
A survey drone has acceleration a→(t)=(2t)i^+6j^\(\overrightarrow{a}\)\(\left\)(t\(\right\))=\(\left\)(2t\(\right\))\(\hat{i}\)+6\(\hat{j}\), initial velocity v→(1)=5i^+2j^\(\overrightarrow{v}\)\(\left\)(1\(\right\))=5\(\hat{i}\)+2\(\hat{j}\), and position r→(1)=4i−j\(\overrightarrow{r}\)\(\left\)(1\(\right\))=4i-j. Which position function satisfies all conditions?
