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

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

Question 1

A vector $$ \vec{A} = 1.50\, \mathrm{m} \; \hat{x} - 2.40\, \mathrm{m} \; \hat{y}  is $$given. What is the magnitude of $$ \vec{A} $$?

Accepted Answer

$$\sqrt{(1.50)^2$ + (-2.40)^2$} = 2.83\, \mathrm{m}$$

Question 2

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

Accepted Answer

$$	heta = 251.1^\circ$$

Question 3

A position vector $$ \vec{r}_i $$ has a magnitude of $$3.50\, \mathrm{m}$$ and points $$145^\circ$$ counterclockwise from the positive x-axis. What is the x-component of $$ \vec{r}_i $$?

Accepted Answer

$$x_i = (3.50) \cos 145^\circ = -2.87\, \mathrm{m}$$

Question 4

An object moves from initial position $$ \vec{r}_i = -2.87\, \mathrm{m} \; \hat{x} + 2.01\, \mathrm{m} \; \hat{y}  to $$final position $$ \vec{r}_f = 2.00\, \mathrm{m} \; \hat{x} - 3.50\, \mathrm{m} \; \hat{y} . $$What is the displacement vector $$ \Delta \vec{r} $$?

Accepted Answer

$$\Delta \vec{r} = 4.87\, \mathrm{m} \; \hat{x} - 5.51\, \mathrm{m} \; \hat{y}$$

Question 5

Which of the following best describes the direction of the instantaneous velocity vector for a particle moving along a curved path?

Accepted Answer

It points tangent to the path at the particle's position.

Question 6

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

Accepted Answer

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

Question 7

Which statement is true about the average velocity vector $$ \vec{v}_{av} $$ for a particle moving from point $$P_1 to$$ $$P_2$$?

Accepted Answer

It points in the same direction as the displacement vector.

Question 8

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

Accepted Answer

$$\vec{a}_{av} = \frac{6.36\, \mathrm{m/s} \; \hat{x} + 8.64\, \mathrm{m/s} \; \hat{y}}{8.00\, \mathrm{s}} = 0.795\, \mathrm{m/s^2$} \; \hat{x} + 1.08\, \mathrm{m/s^2$} \; \hat{y}$$

Question 9

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 10

A particle moves such that its velocity vector always points tangent to its path. What does this imply about the direction of the acceleration vector?

Accepted Answer

The acceleration vector can point in a direction different from the velocity vector.

Question 11

If an object is in free fall and moving upward, what is the direction of its velocity and acceleration vectors?

Accepted Answer

Velocity is upward, acceleration is downward.

Question 12

Which of the following best describes the average acceleration vector $$ \vec{a}_{av} $$?

Accepted Answer

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

Question 13

A vector $$ \vec{C} $$ has components $$C_x = 3.00\, \mathrm{m}$$ and $$C_y = 4.00\, \mathrm{m}. $$What is the magnitude and direction of $$ \vec{C} $$ measured from the positive x-axis?

Accepted Answer

$$C = 5.00\, \mathrm{m}, 	heta = 53.1^\circ$$

Question 14

Which of the following is the correct expression for the displacement vector $$ \Delta \vec{r}  in $$terms of initial and final position vectors?

Accepted Answer

$$\Delta \vec{r} = \vec{r}_f - \vec{r}_i$$

Question 15

A particle moves from $$(2.0\, \mathrm{m}, 3.0\, \mathrm{m}) to$$ $$(5.0\, \mathrm{m}, 7.0\, \mathrm{m}) in$$ $$2.0\, \mathrm{s}. $$What is the magnitude of its average velocity?

Accepted Answer

$$\frac{\sqrt{(5.0-2.0)^2$ + (7.0-3.0)^2$}}{2.0} = 2.5\, \mathrm{m/s}$$

Vectors in Physics: Magnitude, Direction, and Applications

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