Q: A soccer player kicks a ball at an angle of $$30^\circ$$ above the horizontal with an initial speed of $$20\ \mathrm{m/s}. $$Neglecting air resistance, what is the initial vertical component of the ball's velocity?

$$v_{y,i} = 20\ \mathrm{m/s} \times \sin 30^\circ = 10\ \mathrm{m/s}$$

Q: A projectile is launched horizontally from a $$20\ \mathrm{m}$$ high cliff with a speed of $$10\ \mathrm{m/s}. $$How long does it take to hit the ground? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 20}{9.8}} \approx 2.02\ \mathrm{s}$$

Q: A ball is thrown with an initial velocity of $$15\ \mathrm{m/s} at an $$angle of $$45^\circ$$ above the horizontal. What is the range of the projectile? (Neglect air resistance, $$g = 9.8\ \mathrm{m/s^2$})$$

$$R = \frac{v_0$^2 \sin 2\theta}{g} = \frac{15^2$ \sin 90^\circ}{9.8} = \frac{225}{9.8} \approx 22.96\ \mathrm{m}$$

Question 1

A soccer player kicks a ball at an angle of $$30^\circ$$ above the horizontal with an initial speed of $$20\ \mathrm{m/s}. $$Neglecting air resistance, what is the initial vertical component of the ball's velocity?

Accepted Answer

$$v_{y,i} = 20\ \mathrm{m/s} 	imes \sin 30^\circ = 10\ \mathrm{m/s}$$

Question 2

A vector $$\vec{A}$$ has 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 hiker walks 4 km east and then 3 km north. What is the magnitude of the net displacement?

Accepted Answer

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

Question 4

A crate slides down a frictionless ramp inclined at $$25^\circ. $$What is the acceleration of the crate along the ramp?

Accepted Answer

$$a = g \sin 25^\circ$$

Question 5

A projectile is launched horizontally from a $$20\ \mathrm{m}$$ high cliff with a speed of $$10\ \mathrm{m/s}. $$How long does it take to hit the ground? (Use $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 	imes 20}{9.8}} \approx 2.02\ \mathrm{s}$$

Question 6

A vector $$\vec{B} is$$ $$5$$ units long and points $$60^\circ$$ above the positive x-axis. What are its x and y components?

Accepted Answer

$$B_x = 5 \cos 60^\circ = 2.5$$, $$B_y = 5 \sin 60^\circ = 4.33$$

Question 7

A car rounds a curve of radius $$50\ \mathrm{m} at a $$constant speed of $$20\ \mathrm{m/s}. $$What is the magnitude of its centripetal acceleration?

Accepted Answer

$$a_c = \frac{v^2$}{r} = \frac{20^2$}{50} = 8\ \mathrm{m/s^2$}$$

Question 8

A ball is thrown with an initial velocity of $$15\ \mathrm{m/s} at an $$angle of $$45^\circ$$ above the horizontal. What is the range of the projectile? (Neglect air resistance, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$R = \frac{v_0$^2 \sin 2	heta}{g} = \frac{15^2$ \sin 90^\circ}{9.8} = \frac{225}{9.8} \approx 22.96\ \mathrm{m}$$

Question 9

A mouse runs straight across a conveyor belt moving at $$3\ \mathrm{m/s}$$ while the mouse runs at $$4\ \mathrm{m/s}$$ relative to the belt. What is the mouse's speed relative to the ground?

Accepted Answer

$$v = \sqrt{3^2$ + 4^2$} = 5\ \mathrm{m/s}$$

Question 10

A skydiver falls vertically at $$35\ \mathrm{m/s}$$ while a wind blows horizontally at $$7\ \mathrm{m/s}. $$What is the magnitude of the skydiver's velocity relative to the ground?

Accepted Answer

$$v = \sqrt{35^2$ + 7^2$} = \sqrt{1225 + 49} = \sqrt{1274} \approx 35.7\ \mathrm{m/s}$$

Question 11

A vector $$\vec{A} is $$multiplied by $$-2. $$What happens to its magnitude and direction?

Accepted Answer

The magnitude is doubled and the direction is reversed.

Question 12

A projectile is launched and lands at the same height. Which of the following statements is true about its vertical velocity at launch and at landing (neglecting air resistance)?

Accepted Answer

The magnitude is the same but the direction is opposite.

Question 13

A car drives up a $$10^\circ$$ slope at $$15\ \mathrm{m/s}$$ for $$10\ \mathrm{s}. $$How much vertical height does it gain?

Accepted Answer

$$\Delta y = v \sin 	heta 	imes t = 15 \sin 10^\circ 	imes 10 \approx 26\ \mathrm{m}$$

Question 14

A plane flies due east at $$500\ \mathrm{km/h}$$ while a wind blows due south at $$100\ \mathrm{km/h}. $$What is the plane's speed relative to the ground?

Accepted Answer

$$v = \sqrt{500^2$ + 100^2$} = \sqrt{260000} \approx 510\ \mathrm{km/h}$$

Question 15

A toy car moves around a circular track at constant speed. If it suddenly doubles its speed, by what factor does its centripetal acceleration change?

Accepted Answer

It increases by a factor of 4.

Question 16

A projectile is launched at $$30^\circ$$ above the horizontal with an initial speed $$v_0. $$What is the expression for the horizontal component of its velocity?

Accepted Answer

$$v_{x} = v_0$ \cos 30^\circ$$

Question 17

A vector $$\vec{A}$$ has components $$A_x = -3$$ and $$A_y = 4. $$What is the direction of $$\vec{A}$$ relative to the positive x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{4}{-3}ight) = 126.9^\circ$$

Vectors and Motion in Two Dimensions: Study Guide

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