Question 1

Let $$\mathbf{r}(t) = \langle t^2$, \sin t, e^t \rangle. $$What is the velocity vector $$\mathbf{v}(t) at $$time $$t$$?

Accepted Answer

$$\langle 2t, \cos t, e^t \rangle$$

Question 2

A particle moves in space with position $$\mathbf{r}(t) = \langle 3\cos t, 3\sin t, 4t \rangle. $$What is the speed of the particle at time $$t$$?

Accepted Answer

$$\sqrt{9 + 16}$$

Question 3

Suppose $$\mathbf{v}(t) = \langle 2t, 3, 0 \rangle$$ and $$\mathbf{p}(0) = \langle 1, 2, 3 \rangle. $$What is the position vector $$\mathbf{p}(t)$$?

Accepted Answer

$$\langle t^2$ + 1, 3t + 2, 3 \rangle$$

Question 4

A particle moves with velocity $$\mathbf{v}(t) = \langle t^2$, 3t \rangle. $$What is its displacement from $$t=0 to$$ $$t=1$$?

Accepted Answer

$$\left\langle \dfrac{1}{3}, \dfrac{3}{2} \right\rangle$$

Question 5

Let $$\mathbf{r}(t) be a $$position vector such that $$|\mathbf{r}(t)| is $$constant for all $$t. $$What can be said about $$\mathbf{r}(t)$$ and its velocity $$\mathbf{v}(t)$$?

Accepted Answer

$$\mathbf{r}(t) \cdot \mathbf{v}(t) = 0$$ for all $$t$$

Question 6

Given $$\mathbf{r}(t) = \langle 3\sin t \cos 2t, 3\sin t \sin 2t, 3\cos t \rangle$$, what is $$|\mathbf{r}(t)|$$?

Accepted Answer

$$3$$

Question 7

A ball is thrown from position $$\langle 0, 0, 1 \rangle$$ with initial velocity $$4\,\mathrm{m/s} in $$the direction of the unit vector $$\langle \frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}} \rangle. $$What is the $$z-$$component of its position at time $$t ($$neglecting air resistance, $$g is $$the gravitational constant)?

Accepted Answer

$$1 + 2\sqrt{2}t - \dfrac{g}{2}t^2$

Question 8

A projectile is launched from the ground with initial speed $$v_0 at an $$angle $$\alpha$$ above the horizontal. What is the formula for the maximum height reached?

Accepted Answer

$$\dfrac{(v_0$ \sin \$$alpha)^2$$}{2g}$$

Question 9

A cannon is fired at $$60^\circ$$ above the horizontal with initial speed $$100 m/s. $$What is the range of the projectile (distance traveled horizontally before hitting the ground)?

Accepted Answer

$$\dfrac{5000\sqrt{3}}{g}$$

Question 10

Suppose $$\mathbf{r}(t) = \langle a\cos t, a\sin t \rangle. $$What is the geometric path traced by $$\mathbf{r}(t)$$?

Accepted Answer

A circle of radius $$a$$ centered at the origin

Question 11

A particle moves in the plane with position $$\mathbf{r}(t) = \langle t, t^2$ \rangle. $$What is the acceleration vector $$\mathbf{a}(t)$$?

Accepted Answer

$$\langle 0, 2 \rangle$$

Question 12

If $$\mathbf{v}(t) = \langle 0, 1, 2t \rangle$$ and $$\mathbf{p}(0) = \langle 1, 0, 0 \rangle$$, what is $$\mathbf{p}(t)$$?

Accepted Answer

$$\langle 1, t, t^2$ \rangle$$

Question 13

A ball is thrown straight up from height $$h$$ with initial velocity $$v_0. $$How long does it take to reach its maximum height?

Accepted Answer

$$\dfrac{v_0$}{g}$$

Question 14

Given $$\mathbf{r}(t) = \langle t, \cos t, \sin t \rangle$$, what is $$\mathbf{r}'(t) \cdot \mathbf{r}(t) at$$ $$t=0$$?

Accepted Answer

$$1$$

Question 15

A particle moves with velocity $$\mathbf{v}(t) = \langle 3, -gt \rangle$$ and initial position $$\mathbf{r}(0) = \langle 0, 0 \rangle. $$What is the $$y-$$component of its position at time $$t$$?

Accepted Answer

$$-\dfrac{g}{2}t^2$

Question 16

Which of the following statements about the indefinite integral of a vector-valued function $$\mathbf{r}(t) is $$correct?

Accepted Answer

$$\int \mathbf{r}(t) dt = \mathbf{R}(t) + \mathbf{C}$$, where $$\mathbf{C} is an $$arbitrary constant vector

Question 17

A projectile is launched from $$\langle 0, 0 \rangle$$ with initial velocity $$\langle v_0$ \cos \alpha, v_0$ \sin \alpha \rangle. $$What is the time of flight until it returns to the ground?

Accepted Answer

$$\dfrac{2v_0 \sin \alpha}{g}$$

Question 18

If $$\mathbf{r}(t)$$ describes a path on a sphere of radius $$R$$ centered at the origin, what is $$\mathbf{r}(t) \cdot \mathbf{r}(t)$$?

Accepted Answer

$$R^2$$

Question 19

A particle moves with acceleration $$\mathbf{a}(t) = \langle 0, 0, -g \rangle$$ and initial velocity $$\mathbf{v}(0) = \langle v_x, 0, v_z \rangle. $$What is the velocity at time $$t$$?

Accepted Answer

$$\langle v_x, 0, v_z - gt \rangle$$

Calc 7 : Motion in Space: Derivatives, Integrals, and Applications in Calculus

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