Q: Given a vector $$\vec{A}$$ with components $$A_x = 4$$ and $$A_y = 3$$, what is the angle $$\theta it $$makes with the x-axis?

$$\theta = \tan$$^{-1}$$\left(\frac{3}{4}\right) \approx 37$$^{\circ}$$ N of E$

Q: A ball is thrown horizontally off a $$1\ \mathrm{m}$$ high table and lands $$2\ \mathrm{m}$$ from the base. What was its initial horizontal velocity?

$$v_{x0} = \frac{2}{\sqrt{2 \times 1 / 9.8}} \approx 4.43\ \mathrm{m/s}$$

Q: A boat can travel at $$10\ \mathrm{m/s} in $$still water. The Hudson River flows south at $$3\ \mathrm{m/s}. $$What direction should the boat head to travel directly east across the river?

$$\theta = \sin$$^{-1}$$\left(\frac{3}{10}\right) \approx 17.5$$^{\circ}$$ N of E$

Question 1

A person walks 4 blocks east and then 3 blocks north. What is the magnitude of their displacement?

Accepted Answer

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

Question 2

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

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{3}{4}ight) \approx 37$$^{\circ}$$ N of E$

Question 3

Which of the following best describes the resultant vector?

Accepted Answer

The sum of two or more vectors.

Question 4

A plane takes off at $$100\ \mathrm{m/s} at$$ $$30^{\circ}$$ above the horizontal. What is the horizontal component of its velocity?

Accepted Answer

$$V_x = 100 \cos 30$$^{\circ}$$ = 86.6\ \mathrm{m/s}$$

Question 5

An ant walks from position $$(1\ \mathrm{m}, 2\ \mathrm{m}) to$$ $$(4\ \mathrm{m}, -2\ \mathrm{m}). $$What is its displacement vector?

Accepted Answer

$$\Delta \vec{r} = (4-1)\hat{i} + (-2-2)\hat{j} = 3\hat{i} - 4\hat{j}$$

Question 6

A ball is thrown horizontally off a $$1\ \mathrm{m}$$ high table and lands $$2\ \mathrm{m}$$ from the base. What was its initial horizontal velocity?

Accepted Answer

$$v_{x0} = \frac{2}{\sqrt{2 	imes 1 / 9.8}} \approx 4.43\ \mathrm{m/s}$$

Question 7

Which statement about instantaneous velocity is correct?

Accepted Answer

It is always tangent to the path.

Question 8

A hummingbird has initial velocity $$(2\ \mathrm{m/s})\hat{i} + (1\ \mathrm{m/s})\hat{j}$$ and after $$0.5\ \mathrm{s}$$ its velocity is $$(-1\ \mathrm{m/s})\hat{i} + (-5\ \mathrm{m/s})\hat{j}. $$What is its average acceleration vector?

Accepted Answer

$$\vec{a} = \frac{(-1-2)\hat{i} + (-5-1)\hat{j}}{0.5} = (-6\hat{i} - 12\hat{j})\ \mathrm{m/s^2$}$$

Question 9

A boat can travel at $$10\ \mathrm{m/s} in $$still water. The Hudson River flows south at $$3\ \mathrm{m/s}. $$What direction should the boat head to travel directly east across the river?

Accepted Answer

$$	heta = \sin$$^{-1}$$\left(\frac{3}{10}ight) \approx 17.5$$^{\circ}$$ N of E$

Question 10

A car is driving north at $$12\ \mathrm{m/s}$$, and a truck is driving east at $$16\ \mathrm{m/s}. $$What is the velocity of the car relative to the truck?

Accepted Answer

$$\vec{v}_{ct} = 12\hat{j} - 16\hat{i}\ \mathrm{m/s}$$

Question 11

A ball in a computer game has horizontal acceleration $$2\ \mathrm{m/s^2$}$$ and vertical acceleration $$3\ \mathrm{m/s^2$}. If it $$starts at rest at the origin, where is it after $$4\ \mathrm{s}$$?

Accepted Answer

$$x = \frac{1}{2} a_x t^2$ = 16\ \mathrm{m},\ y = \frac{1}{2} a_y t^2$ = 24\ \mathrm{m}$$

Question 12

Which equation gives the magnitude of a vector $$\vec{A}$$ with components $$A_x$$ and $$A_y$$?

Accepted Answer

$$|\vec{A}| = \sqrt{A_x^2$ + A_y^2$}$$

Question 13

If $$\vec{A} = 5\hat{i} + 12\hat{j}$$, what is $$|\vec{A}|$$?

Accepted Answer

$$|\vec{A}| = \sqrt{5^2$ + 12^2$} = 13$$

Question 14

Which of the following is true about vector addition?

Accepted Answer

Vector addition is commutative: $$\vec{A} + \vec{B} = \vec{B} + \vec{A}$$

Question 15

A car moves at $$25\ \mathrm{m/s} at$$ $$30^{\circ} N of E. $$What is its x-displacement after $$10\ \mathrm{s}$$?

Accepted Answer

$$\Delta x = v_x t = (25 \cos 30$$^{\circ}$$) 	imes 10 = 216.5\ \mathrm{m}$$

Question 16

A rock is thrown horizontally off a $$100\ \mathrm{m}$$ cliff. How long does it take to hit the ground? (Ignore air resistance)

Accepted Answer

$$t = \sqrt{\frac{2 	imes 100}{9.8}} \approx 4.52\ \mathrm{s}$$

Question 17

Which of the following best defines a reference frame?

Accepted Answer

An environment in which you perform an experiment or carry out a measurement.

Question 18

If your velocity relative to the earth is $$+1\ \mathrm{m/s} ($$east), what is the earth's velocity relative to you?

Accepted Answer

$$-1\ \mathrm{m/s} ($$west)

Vectors and Two-Dimensional Kinematics: Study Notes

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