Q: A vector $$\vec{C}$$ has components $$C_x = -6$$ and $$C_y = 8. $$What is the direction angle $$\theta ($$relative to the $$x-$$axis) of $$\vec{C}$$?

$$\theta = \arctan\left(\frac{8}{-6}\right)$$

Question 1

Which of the following best describes a vector?

Accepted Answer

A quantity that has both magnitude and direction.

Question 2

Given a vector $$\vec{A}$$ with components $$A_x = 3$$ and $$A_y = 4$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$A = \sqrt{3^2$ + 4^2$} = 5$$

Question 3

A displacement of $$5\ \mathrm{km} is $$directed $$53^\circ$$ east of south. What is the $$x-$$component (east) of the displacement?

Accepted Answer

$$D_x = 5 \sin 53^\circ = 4\ \mathrm{km}$$

Question 4

Which equation correctly expresses a vector $$\vec{A} in $$unit vector notation if $$A_x = A \cos 	heta$$ and $$A_y = A \sin 	heta$$?

Accepted Answer

$$\vec{A} = A_x \hat{i} + A_y \hat{j}$$

Question 5

If $$\vec{A} = 3\hat{i} + 4\hat{j}$$ and $$\vec{B} = -2\hat{i} + 5\hat{j}$$, what is $$\vec{C} = \vec{A} + \vec{B}$$?

Accepted Answer

$$\vec{C} = (3 + (-2))\hat{i} + (4 + 5)\hat{j} = 1\hat{i} + 9\hat{j}$$

Question 6

A particle moves such that its position vector is $$\vec{r}(t) = (2t)\hat{i} + (3t^2$)\hat{j}. $$What is its velocity vector $$\vec{v}(t)$$?

Accepted Answer

$$\vec{v}(t) = \frac{d\vec{r}}{dt} = 2\hat{i} + 6t\hat{j}$$

Question 7

Which of the following statements about acceleration is correct?

Accepted Answer

Acceleration can change the magnitude and/or direction of velocity.

Question 8

A ball is dropped from rest and falls freely under gravity. What is the $$y-$$component of its velocity after $$t$$ seconds?

Accepted Answer

$$v_y = -gt$$

Question 9

A car is moving east at $$10\ \mathrm{m/s}. A $$ball is launched vertically upward from the car with a velocity of $$15\ \mathrm{m/s}. $$What is the initial velocity vector of the ball in unit vector notation?

Accepted Answer

$$\vec{v}_0 = 10\hat{i} + 15\hat{j}\ \mathrm{m/s}$$

Question 10

If $$\vec{A} = 5\hat{i} + 2\hat{j}$$ and $$\vec{B} = 3\hat{i} - 4\hat{j}$$, what is $$\vec{D} = \vec{A} - \vec{B}$$?

Accepted Answer

$$\vec{D} = (5 - 3)\hat{i} + (2 - (-4))\hat{j} = 2\hat{i} + 6\hat{j}$$

Question 11

A projectile is launched horizontally from a cliff with initial velocity $$v_0$$ and height $$h. $$Neglecting air resistance, what is the $$y-$$component of its position after time $$t$$?

Accepted Answer

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

Question 12

Which of the following is true about the separation of motion in two dimensions?

Accepted Answer

The $$x$$ and $$y$$ components of motion are independent except for the time variable.

Question 13

A vector $$\vec{C}$$ has components $$C_x = -6$$ and $$C_y = 8. $$What is the direction angle $$	heta ($$relative to the $$x-$$axis) of $$\vec{C}$$?

Accepted Answer

$$	heta = \arctan\left(\frac{8}{-6}ight)$$

Question 14

If a ball is simultaneously dropped and another is launched horizontally from the same height, which hits the ground first (neglecting air resistance)?

Accepted Answer

Both hit the ground at the same time.

Question 15

Which of the following is the correct formula for the instantaneous velocity vector $$\vec{v} in $$terms of the position vector $$\vec{r}$$?

Accepted Answer

$$\vec{v} = \frac{d\vec{r}}{dt}$$

Question 16

A vector $$\vec{A}$$ makes an angle $$	heta$$ with the $$x-$$axis. Which of the following gives the $$y-$$component of $$\vec{A}$$?

Accepted Answer

$$A_y = A \sin 	heta$$

Question 17

Which of the following best describes the effect of an acceleration component perpendicular to the velocity vector?

Accepted Answer

It changes the direction of the velocity vector.

Vectors and Two-Dimensional Kinematics

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