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

Which of the following best describes a vector quantity?

Accepted Answer

A quantity with magnitude and direction, such as velocity.

Question 2

A hiker walks 3 km east and then 4 km north. What is the magnitude of the hiker's displacement?

Accepted Answer

$$5\ \mathrm{km}$$

Question 3

Given a velocity vector $$\vec{v} = 3\hat{i} + 4\hat{j} m/s$$, what is the direction of the velocity relative to the positive x-axis?

Accepted Answer

$$53.1^\circ$$

Question 4

A projectile is launched with an initial speed $$v_0 at an $$angle $$\$$alpha_0$$ above the horizontal. Which equation gives the horizontal position x$ as a function of time t$?

Accepted Answer

$$x = v_0$ \cos \$$alpha_0$$\ t$$

Question 5

Which of the following statements about instantaneous velocity is correct?

Accepted Answer

It is always tangent to the path of the particle.

Question 6

A car moves in a circle of radius $$12.0\ \mathrm{m}$$, completing one revolution every $$6.00\ \mathrm{s}. $$What is the magnitude of its average acceleration after a quarter revolution?

Accepted Answer

$$\frac{2v^2$}{\pi r}$$

Question 7

A river is $$25.0\ \mathrm{m}$$ wide and flows east at $$1.00\ \mathrm{m/s}. A $$boat can move at $$2.00\ \mathrm{m/s}$$ relative to the water. If the boat heads directly north, how far downstream will it land when it reaches the opposite bank?

Accepted Answer

$$12.5\ \mathrm{m}$$

Question 8

Which of the following is the correct expression for the y-component of a projectile's velocity at time $$t$$?

Accepted Answer

$$v_y = v_0$ \sin \$$alpha_0$$ - gt$$

Question 9

A pilot wants to fly due north with an airspeed of $$147\ \mathrm{m/s}$$, but there is a wind blowing east at $$24\ \mathrm{m/s}. At $$what angle west of north should the pilot head?

Accepted Answer

$$\arctan\left(\frac{24}{147}\right)$$

Question 10

Which of the following is the correct definition of a unit vector $$\hat{r}$$?

Accepted Answer

$$\hat{r} = \frac{\vec{r}}{|\vec{r}|}$$

Question 11

A bug moves from position $$\vec{r}_1 = 4.0\hat{i} - 3.0\hat{j} to$$ $$\vec{r}_2 = 3.0\hat{i} + 1.0\hat{j} in$$ $$4.0\ \mathrm{s}. $$What is the average velocity vector?

Accepted Answer

$$\vec{v}_{avg} = (-0.25\hat{i} + 1.0\hat{j})\ \mathrm{m/s}$$

Question 12

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 13

A ferry boat heads at $$35^\circ$$ north of east across a river with a current flowing east. If the boat's speed relative to the water is $$2.5\ \mathrm{m/s}$$ and the current is $$1.0\ \mathrm{m/s}$$, what is the boat's speed relative to the shore?

Accepted Answer

$$\sqrt{(2.5\sin 35^\$$circ)^2$$ + (2.5\cos 35^\circ + 1.0)^2$}$$

Question 14

Which of the following best describes the process for solving relative motion problems in two dimensions?

Accepted Answer

Identify all three objects, write down knowns, relate velocities using $$\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$$, and flip subscripts if necessary.

Question 15

A cyclist rides due east at $$5.0\ \mathrm{m/s}$$ while a wind blows due north at $$3.0\ \mathrm{m/s}. $$What is the magnitude of the cyclist's velocity relative to the ground?

Accepted Answer

$$\sqrt{5.0^2$ + 3.0^2$} = 5.83\ \mathrm{m/s}$$

Question 16

If a vector $$\vec{A}$$ has components $$A_x = -3$$ and $$A_y = 4$$, what is the angle it makes with the positive x-axis?

Accepted Answer

$$126.9^\circ$$

Question 17

A ball is thrown horizontally from a cliff with speed $$v_0. $$Which equation gives its vertical position $$y$$ after time $$t ($$assuming $$y=0 at $$the top)?

Accepted Answer

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

Question 18

Which of the following is NOT a correct statement about the components of a vector in 2D?

Accepted Answer

The x-component is always the sine of the angle.

Question 19

A person walks 10 m north, then 10 m east, then 10 m south. What is the magnitude of their displacement from the starting point?

Accepted Answer

$$10\ \mathrm{m}$$

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

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