Q: A projectile is launched from the ground with an initial velocity $$v_0 at an $$angle $$\theta$$ above the horizontal. Neglecting air resistance, what is the time of flight until it returns to the ground?

$$t = \frac{2 v_0$ \sin \theta}{g}$$

Question 1

A particle moves from point $$A at $$position vector $$\vec{r}_1 = 2\hat{i} - 4\hat{j} + 8\hat{k} to $$point $$B at$$ $$\vec{r}_2 = 6\hat{i} + 0\hat{j} + 3\hat{k}. $$What is the displacement vector $$\Delta \vec{r}$$ from $$A to$$ $$B$$?

Accepted Answer

$$4\hat{i} + 4\hat{j} - 5\hat{k}$$

Question 2

Which of the following best describes the average velocity vector $$\vec{v}_{avg}$$ between two points in space?

Accepted Answer

$$\vec{v}_{avg} = \frac{\Delta \vec{r}}{\Delta t}$$

Question 3

A projectile is launched from the ground with an initial velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. Neglecting air resistance, what is the time of flight until it returns to the ground?

Accepted Answer

$$t = \frac{2 v_0$ \sin 	heta}{g}$$

Question 4

A cyclist moves in a circular path of radius $$R at $$constant speed $$v. $$What is the magnitude of the acceleration?

Accepted Answer

$$a = \frac{v^2$}{R}$$

Question 5

A ball is thrown horizontally from a cliff with speed $$v_0. $$Which equation gives its vertical position $$y as a $$function of time $$t$$, assuming $$y=0 at $$the top and downward is negative?

Accepted Answer

$$y = -\frac{1}{2}gt^2$

Question 6

Which of the following statements about projectile motion (neglecting air resistance) is correct?

Accepted Answer

The horizontal and vertical motions are independent except for sharing the same time of flight.

Question 7

A car moves such that its position as a function of time is $$\vec{r}(t) = (3t^2$)\hat{i} + (2t)\hat{j}. $$What is the instantaneous velocity at $$t=2 s$$?

Accepted Answer

$$\vec{v} = (12)\hat{i} + (2)\hat{j}$$

Question 8

A projectile is launched at $$v_0 = 20$ m/s at 30$$^\circ$$ above the horizontal. What is the maximum height reached? (Use g = 9.8$ m/s^2$)

Accepted Answer

$$h_{max} = \frac{(v_0$ \sin \$$theta)^2$$}{2g}$$

Question 9

A particle's velocity is given by $$\vec{v}(t) = (4t)\hat{i} + (3)\hat{j}. $$What is the acceleration vector?

Accepted Answer

$$\vec{a} = 4\hat{i}$$

Question 10

A baseball is hit so that its initial velocity is $$v_0 = 40$ m/s at 37$$^\circ$$ above the horizontal. What is the horizontal component of its velocity?

Accepted Answer

$$v_{0x} = v_0$ \cos 	heta$$

Question 11

A drone flies with velocity $$\vec{v}_{rel,air} = 10\hat{i} m/s $$relative to the air, and the wind blows with $$\vec{v}_{air,ground} = 5\hat{j} m/s. $$What is the drone's velocity relative to the ground?

Accepted Answer

$$\vec{v}_{rel,ground} = 10\hat{i} + 5\hat{j}$$

Question 12

Which of the following is the correct expression for the unit vector $$\hat{u} in $$the direction of a vector $$\vec{A} = A_x \hat{i} + A_y \hat{j} + A_z \hat{k}$$?

Accepted Answer

$$\hat{u} = \frac{\vec{A}}{\sqrt{A_x^2$ + A_y^2$ + A_z^2$}}$$

Question 13

A projectile is launched from a height $$h$$ above the ground with horizontal velocity $$v_0. $$How long does it take to hit the ground?

Accepted Answer

$$t = \sqrt{\frac{2h}{g}}$$

Question 14

A particle moves such that $$x(t) = 2t^2$, y(t) = 3t$, and z(t) = 4$. What is the magnitude of its velocity at t=1$ s?

Accepted Answer

$$|\vec{v}| = \sqrt{(4)^2$ + (3)^2$ + (0)^2$} = 5 m/s$$

Question 15

A runner completes a lap around a 400 m circular track in 50 s. What is the magnitude of their average velocity?

Accepted Answer

0 m/s

Question 16

Which of the following is true about the direction of instantaneous velocity?

Accepted Answer

It is tangent to the particle's path at that instant.

Question 17

A projectile is launched at $$v_0$$ and angle $$	heta. $$What is the range $$R on $$level ground?

Accepted Answer

$$R = \frac{v_0$^2 \sin 2	heta}{g}$$

Question 18

A particle's acceleration vector is always perpendicular to its velocity vector. What type of motion does this describe?

Accepted Answer

Uniform circular motion

Motion in Two and Three Dimensions: Vectors, Velocity, and Acceleration

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