Q: A projectile is launched with initial speed $$v_0 at an $$angle $$\theta$$ above the horizontal. What is the maximum height $$h$$ reached by the projectile?

$$h = \dfrac{v_0$^2 \sin^2$ \theta}{2g}$$

Question 1

Given the position vector $$\vec{r}(t) = \langle t^2$, \sin t, e^t \rangle$$, what is the velocity vector $$\vec{v}(t) at $$time $$t$$?

Accepted Answer

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

Question 2

A particle moves along a path described by $$\vec{r}(t) = \langle \cos t, \sin t, t \rangle. $$What is the speed of the particle at $$t = \pi$$?

Accepted Answer

$$\sqrt{2}$$

Question 3

Which of the following is the correct formula for the unit tangent vector $$\vec{T}(t) of a $$curve with position vector $$\vec{r}(t)$$?

Accepted Answer

$$\vec{T}(t) = \dfrac{\vec{r}'(t)}{\|\vec{r}'(t)\|}$$

Question 4

Given the curve $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle$$, find the acceleration vector $$\vec{a}(t).$$

Accepted Answer

$$\langle 0, 2, 6t \rangle$$

Question 5

A particle moves along the curve $$\vec{r}(t) = \langle 3\cos t, 3\sin t, t \rangle. $$What is the magnitude of the velocity at $$t = 0$$?

Accepted Answer

$$\sqrt{9 + 1}$$

Question 6

Which of the following best describes the curvature $$\kappa of a $$curve at a point?

Accepted Answer

$$\kappa = \dfrac{\|\vec{T}'(t)\|}{\|\vec{r}'(t)\|}$$

Question 7

A particle moves along a helix $$\vec{r}(t) = \langle \cos t, \sin t, t \rangle. $$What is the equation of the tangent line at $$t = 0$$?

Accepted Answer

$$\vec{L}(s) = \langle 1, 0, 0 \rangle + s \langle 0, 1, 1 \rangle$$

Question 8

If $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle$$, what is the equation of the normal plane at $$t = 1$$?

Accepted Answer

$$x + 2y + 3z = 6$$

Question 9

A particle moves along $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle. At$$ $$t = 2$$, what is the binormal vector $$\vec{B}(t)$$?

Accepted Answer

$$\langle -1, 2, -1 \rangle$$

Question 10

Given the curve $$\vec{r}(t) = \langle 2t, t^2$, t \rangle$$, find the equation of the osculating plane at $$t = 1.$$

Accepted Answer

$$2x + y + z = 5$$

Question 11

A disk of radius $$r$$ rolls along the $$x-$$axis without slipping. If the center moves at constant speed $$v$$, what is the angular velocity $$\omega of $$the disk?

Accepted Answer

$$\omega = \dfrac{v}{r}$$

Question 12

A projectile is launched with initial speed $$v_0 at an $$angle $$	heta$$ above the horizontal. What is the maximum height $$h$$ reached by the projectile?

Accepted Answer

$$h = \dfrac{v_0$^2 \sin^2$ 	heta}{2g}$$

Question 13

A particle moves with position function $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle. $$What is the speed at $$t = 1$$?

Accepted Answer

$$\sqrt{1^2$ + (2 \cdot 1)^2$ + (3 \cdot 1^2$)^2$$}$

Question 14

Find the parametric equations for the tangent line to the curve $$\vec{r}(t) = \langle t^2$, t^3$, t \rangle at$$ $$t = 1.$$

Accepted Answer

$$x = 1 + 2s,\ y = 3 + 3s,\ z = 1 + s$$

Question 15

A curve is given by $$\vec{r}(t) = \langle \cos t, \sin t, t \rangle. $$What is the arc length from $$t = 0 to$$ $$t = 2\pi$$?

Accepted Answer

$$2\pi \sqrt{2}$$

Question 16

Given $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle$$, what is the equation of the tangent plane at $$t = 1$$?

Accepted Answer

$$x + 2y + 3z = 6$$

Question 17

A particle moves along $$\vec{r}(t) = \langle t, t^2$, t^3$ \rangle. $$What is the acceleration at $$t = 2$$?

Accepted Answer

$$\langle 0, 2, 12 \rangle$$

Question 18

Find the equation of the osculating circle for the curve $$\vec{r}(t) = \langle \cos t, \sin t \rangle at$$ $$t = 0.$$

Accepted Answer

$$(x - 1)^2$ + y^2$ = 1$$

Study Notes: Vector Calculus and Kinematics in Physics

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