Q: A vector $$ \vec{A} = 3.00\,\text{m} \; \hat{x} + 4.00\,\text{m} \; \hat{y} is $$given. What is the magnitude of $$ \vec{A} $$?

$$\sqrt{3.00^2$ + 4.00^2$} = 5.00\,\text{m}$$

Q: A car changes its velocity from $$ \vec{v}_i = -6.36\,\text{m/s} \hat{x} + 6.36\,\text{m/s} \hat{y} to$$ $$ \vec{v}_f = 15.00\,\text{m/s} \hat{y} in 8.00 s. $$What is the magnitude of the average acceleration?

$$a_{av} = \sqrt{0.795^2$ + 1.08^2$} = 1.34\,\text{m/s}^2$$

Q: Which of the following is the correct formula for the instantaneous acceleration vector?

$$\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta \vec{v}}{\Delta t}$$

Q: A vector $$ \vec{D} $$ has a magnitude of 5.00 m and points at 120° counterclockwise from the positive x-axis. What are its x and y components?

$$D_x = 5.00 \cos 120^\circ = -2.50\,\text{m},\; D_y = 5.00 \sin 120^\circ = 4.33\,\text{m}$$

Question 1

A vector $$ \vec{A} = 3.00\,	ext{m} \; \hat{x} + 4.00\,	ext{m} \; \hat{y}  is $$given. What is the magnitude of $$ \vec{A} $$?

Accepted Answer

$$\sqrt{3.00^2$ + 4.00^2$} = 5.00\,	ext{m}$$

Question 2

A position vector is given by $$ \vec{r} = x \hat{x} + y \hat{y} . If an $$object moves from $$ \vec{r}_i = 2.0\,	ext{m} \hat{x} + 3.0\,	ext{m} \hat{y}  to$$ $$ \vec{r}_f = 5.0\,	ext{m} \hat{x} + 7.0\,	ext{m} \hat{y} $$, what is the displacement vector $$ \Delta \vec{r} $$?

Accepted Answer

$$\Delta \vec{r} = (5.0 - 2.0)\,\hat{x} + (7.0 - 3.0)\,\hat{y} = 3.0\,\hat{x} + 4.0\,\hat{y}$$

Question 3

Given a vector $$ \vec{B} = -1.20\,	ext{m} \hat{x} - 3.50\,	ext{m} \hat{y} $$, what is the direction of $$ \vec{B} $$ measured counterclockwise from the positive x-axis?

Accepted Answer

$$	heta = 251.1^\circ$$

Question 4

A dragonfly moves from $$ \vec{r}_i = 4.60\,	ext{m} \hat{x} + 0.960\,	ext{m} \hat{y}  to$$ $$ \vec{r}_f = 0.520\,	ext{m} \hat{x} + 2.40\,	ext{m} \hat{y}  in 3.00 s. $$What is the average velocity vector $$ \vec{v}_{av} $$?

Accepted Answer

$$\vec{v}_{av} = \frac{-4.08\,\hat{x} + 1.44\,\hat{y}}{3.00} = -1.36\,	ext{m/s} \hat{x} + 0.480\,	ext{m/s} \hat{y}$$

Question 5

Which of the following statements about the instantaneous velocity vector is correct?

Accepted Answer

It points in the direction of motion at any given instant.

Question 6

A car changes its velocity from $$ \vec{v}_i = -6.36\,	ext{m/s} \hat{x} + 6.36\,	ext{m/s} \hat{y}  to$$ $$ \vec{v}_f = 15.00\,	ext{m/s} \hat{y}  in 8.00 s. $$What is the magnitude of the average acceleration?

Accepted Answer

$$a_{av} = \sqrt{0.795^2$ + 1.08^2$} = 1.34\,	ext{m/s}^2$$

Question 7

Which of the following best describes the direction of the average acceleration vector?

Accepted Answer

It points in the direction of the change in velocity vector.

Question 8

If a vector $$ \vec{C} $$ has components $$ C_x = -2.00\,	ext{m} $$ and $$ C_y = 2.00\,	ext{m} $$, what is the angle $$ 	heta  ($$measured counterclockwise from the positive x-axis) of $$ \vec{C} $$?

Accepted Answer

$$	heta = 135^\circ$$

Question 9

An object in free fall is moving upward. Which statement is true about its velocity and acceleration vectors?

Accepted Answer

The velocity vector points upward, and the acceleration vector points downward.

Question 10

A particle moves such that its position vector changes from $$ \vec{r}_i = 1.0\,	ext{m} \hat{x} + 2.0\,	ext{m} \hat{y}  to$$ $$ \vec{r}_f = 4.0\,	ext{m} \hat{x} + 6.0\,	ext{m} \hat{y}  in 2.0 s. $$What is the magnitude of its average velocity?

Accepted Answer

$$v_{av} = \frac{\sqrt{(4.0-1.0)^2$ + (6.0-2.0)^2$}}{2.0} = \frac{5.0}{2.0} = 2.5\,	ext{m/s}$$

Question 11

Which of the following is the correct formula for the instantaneous acceleration vector?

Accepted Answer

$$\vec{a} = \lim_{\Delta t 	o 0} \frac{\Delta \vec{v}}{\Delta t}$$

Question 12

A vector $$ \vec{D} $$ has a magnitude of 5.00 m and points at 120° counterclockwise from the positive x-axis. What are its x and y components?

Accepted Answer

$$D_x = 5.00 \cos 120^\circ = -2.50\,	ext{m},\; D_y = 5.00 \sin 120^\circ = 4.33\,	ext{m}$$

Question 13

Which of the following best describes the relationship between the average velocity vector and the displacement vector?

Accepted Answer

The average velocity vector points in the same direction as the displacement vector.

Question 14

A particle's velocity changes from $$ \vec{v}_i = 2.0\,	ext{m/s} \hat{x}  to$$ $$ \vec{v}_f = 2.0\,	ext{m/s} \hat{y}  in 4.0 s. $$What is the average acceleration vector?

Accepted Answer

$$\vec{a}_{av} = \frac{\vec{v}_f - \vec{v}_i}{4.0} = \frac{-2.0\,\hat{x} + 2.0\,\hat{y}}{4.0} = -0.5\,\hat{x} + 0.5\,\hat{y}\,	ext{m/s}^2$$

Question 15

Which of the following is a correct statement about the acceleration vector for an object in projectile motion?

Accepted Answer

The acceleration vector always points downward due to gravity.

Vectors in Physics: Magnitude, Direction, and Applications

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