Q: A ball is thrown horizontally from the top of a 45 m high cliff with an initial speed of 10 m/s. How far from the base of the cliff will the ball land? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

$$ \text{First, find the time to fall:} y$$ = $$y_0 + v$$_{y0} t + \frac{1}{2} a $$t^2$$ $$0 = 45 + 0 \cdot t - \frac{1}{2} (9.8) t^2$ t$$ = \sqrt{\frac{2 \cdot 45}{9.8}} \approx 3.03\ \mathrm{s}$$ \text{Horizontal distance:} x = v$$_{x0} t = 10 \cdot 3.03 = 30.3\ \mathrm{m}$$ $$ $$30.3\ \mathrm{m}$$

Q: A projectile is launched from the ground at an angle of $$37^\circ$$ above the horizontal with an initial speed of $$20\ \mathrm{m/s}. $$What is the maximum height reached by the projectile? (Use $$\sin 37^\circ = 0.6$$, $$g = 9.8\ \mathrm{m/s^2$})$$

$$ \text{Vertical component: } v_{y0} = 20 \cdot 0.6 = 12\ \mathrm{m/s}$$ \text{Max height: } h = \frac{v_{y0}^2}{2g} = \frac{$$12^2$$}{2 \cdot 9.8} = \frac{144}{19.6} \approx 7.35\ \mathrm{m}$$ $$ $$7.35\ \mathrm{m}$$

Q: A car travels around a circular track of radius $$50\ \mathrm{m} at a $$constant speed of $$20\ \mathrm{m/s}. $$What is the magnitude of its centripetal acceleration?

$$ \text{Centripetal acceleration: } a_c = \frac{v^2$}{r} = \frac{20^2$}{50} = \frac{400}{50} = 8\ \mathrm{m/s^2$}$$ $$ 8$$\ \mathrm{$$m/s^2$$}$

Q: A stone is thrown with an initial velocity of $$15\ \mathrm{m/s} at an $$angle of $$53^\circ$$ above the horizontal from the edge of a $$20\ \mathrm{m}$$ high cliff. How long does it take to hit the ground? ($$\sin 53^\circ = 0.8$$, $$\cos 53^\circ = 0.6$$, $$g = 9.8\ \mathrm{m/s^2$})$$

$$ \text{Vertical component: } v_{y0} = 15 \cdot 0.8 = 12\ \mathrm{m/s}$$ \text{Use: } y = $$y_0 + v$$_{y0} t - \frac{1}{2} g $$t^2$$ $$0 = 20 + 12 t - 4.9 t^2$ 4.9$$ $$t^2 - 12 t - 20 = 0$ \text{Quadratic formula: } t = \frac{12 + \sqrt{12^2$ + 4 \cdot 4.9 \cdot 20}}{2 \cdot 4.9}$$ $$= \frac{12 + \sqrt{144 + 392}}{9.8} = \frac{12 + \sqrt{536}}{9.8}$$ $$\sqrt{536} \approx 23.15$$ $$t = \frac{12 + 23.15}{9.8} \approx \frac{35.15}{9.8} \approx 3.59\ \mathrm{s}$$ $$ 3.59$$\ \mathrm{s}$$

Q: A projectile is launched at $$45^\circ$$ with an initial speed $$v_0. $$What is the ratio of its maximum height to its range? (Neglect air resistance)

$$ \text{Max height: } h = \frac{v_0$^2 \sin^2$ \theta}{2g}$$ \text{Range: } R = \frac{$$v_0^2$$ \sin 2\theta}{g}$$ \text{For } \theta = 45^\circ, \sin 2\theta = \sin 90^\circ = 1$$ $$\sin^2$ 45^\circ = (\frac{\sqrt{2}}{2})^2$ = 0.5$$ $$\text{Ratio: } \frac{h}{R} = \frac{\frac{v_0$^2 \cdot 0.5}{2g}}{\frac{v_0$^2 \cdot 1}{g}} = \frac{0.5}{2} = 0.25$$ $$ 0.25$

Q: A race car travels around a circular track of radius $$r at $$speed $$v. If $$the radius is doubled and the speed is doubled, what happens to the centripetal acceleration?

$$ \text{Original: } a_{c1} = \frac{v^2$}{r}$$ \text{New: } a_{c2} = \frac{$$(2v)^2$$}{2r} = \frac{$$4v^2$$}{2r} = 2 \frac{$$v^2$$}{r}$$ $$ $It doubles.

Q: A projectile is launched from a height $$h$$ above the ground with initial velocity $$v_0 at an $$angle $$\theta$$ above the horizontal. Which equation gives the total time in the air?

$$t = \frac{v_0$ \sin \theta + \sqrt{(v_0$ \sin \$$theta)^2$$ + 2gh}}{g}$$

Question 1

A ball is thrown horizontally from the top of a 45 m high cliff with an initial speed of 10 m/s. How far from the base of the cliff will the ball land? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$
	ext{First, find the time to fall:}

y$$ = $$y_0 + v$$_{y0} t + \frac{1}{2} a $$t^2$$

$$0 = 45 + 0 \cdot t - \frac{1}{2} (9.8) t^2$

t$$ = \sqrt{\frac{2 \cdot 45}{9.8}} \approx 3.03\ \mathrm{s}$$

ext{Horizontal distance:}

x = v$$_{x0} t = 10 \cdot 3.03 = 30.3\ \mathrm{m}$$
$$
$$30.3\ \mathrm{m}$$

Question 2

A projectile is launched from the ground at an angle of $$37^\circ$$ above the horizontal with an initial speed of $$20\ \mathrm{m/s}. $$What is the maximum height reached by the projectile? (Use $$\sin 37^\circ = 0.6$$, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$
	ext{Vertical component: } v_{y0} = 20 \cdot 0.6 = 12\ \mathrm{m/s}$$

ext{Max height: } h = \frac{v_{y0}^2}{2g} = \frac{$$12^2$$}{2 \cdot 9.8} = \frac{144}{19.6} \approx 7.35\ \mathrm{m}$$
$$
$$7.35\ \mathrm{m}$$

Question 3

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

Accepted Answer

$$
	ext{Centripetal acceleration: } a_c = \frac{v^2$}{r} = \frac{20^2$}{50} = \frac{400}{50} = 8\ \mathrm{m/s^2$}$$
$$
8$$\ \mathrm{$$m/s^2$$}$

Question 4

A stone is thrown with an initial velocity of $$15\ \mathrm{m/s} at an $$angle of $$53^\circ$$ above the horizontal from the edge of a $$20\ \mathrm{m}$$ high cliff. How long does it take to hit the ground? ($$\sin 53^\circ = 0.8$$, $$\cos 53^\circ = 0.6$$, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$
	ext{Vertical component: } v_{y0} = 15 \cdot 0.8 = 12\ \mathrm{m/s}$$

ext{Use: } y = $$y_0 + v$$_{y0} t - \frac{1}{2} g $$t^2$$

$$0 = 20 + 12 t - 4.9 t^2$

4.9$$ $$t^2 - 12 t - 20 = 0$

ext{Quadratic formula: } t = \frac{12 + \sqrt{12^2$ + 4 \cdot 4.9 \cdot 20}}{2 \cdot 4.9}$$

$$= \frac{12 + \sqrt{144 + 392}}{9.8} = \frac{12 + \sqrt{536}}{9.8}$$

$$\sqrt{536} \approx 23.15$$

$$t = \frac{12 + 23.15}{9.8} \approx \frac{35.15}{9.8} \approx 3.59\ \mathrm{s}$$
$$
3.59$$\ \mathrm{s}$$

Question 5

A cyclist rides at $$8\ \mathrm{m/s}$$ and throws a ball forward at $$12\ \mathrm{m/s}$$ relative to herself. What is the speed of the ball relative to the ground?

Accepted Answer

$$20\ \mathrm{m/s}$$

Question 6

A car is moving in a circle at constant speed. Which of the following statements is true about its acceleration?

Accepted Answer

The acceleration is directed toward the center of the circle and changes only the direction of velocity.

Question 7

A projectile is launched at $$45^\circ$$ with an initial speed $$v_0. $$What is the ratio of its maximum height to its range? (Neglect air resistance)

Accepted Answer

$$
	ext{Max height: } h = \frac{v_0$^2 \sin^2$ 	heta}{2g}$$

ext{Range: } R = \frac{$$v_0^2$$ \sin 2	heta}{g}$$

ext{For } 	heta = 45^\circ, \sin 2	heta = \sin 90^\circ = 1$$

$$\sin^2$ 45^\circ = (\frac{\sqrt{2}}{2})^2$ = 0.5$$

$$	ext{Ratio: } \frac{h}{R} = \frac{\frac{v_0$^2 \cdot 0.5}{2g}}{\frac{v_0$^2 \cdot 1}{g}} = \frac{0.5}{2} = 0.25$$
$$
0.25$

Question 8

A race car travels around a circular track of radius $$r at $$speed $$v. If $$the radius is doubled and the speed is doubled, what happens to the centripetal acceleration?

Accepted Answer

$$
	ext{Original: } a_{c1} = \frac{v^2$}{r}$$
	ext{New: } a_{c2} = \frac{$$(2v)^2$$}{2r} = \frac{$$4v^2$$}{2r} = 2 \frac{$$v^2$$}{r}$$
$$
$It doubles.

Question 9

A ball is thrown at $$30^\circ$$ above the horizontal with a speed of $$25\ \mathrm{m/s}. $$What is the horizontal component of its velocity?

Accepted Answer

$$
\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866$$

$$v_x = 25 \cdot 0.866 = 21.65\ \mathrm{m/s}$$
$$
21.65$$\ \mathrm{m/s}$

Question 10

A satellite orbits Earth in a circular path at constant speed. Which of the following best describes the direction of its velocity and acceleration vectors?

Accepted Answer

Velocity is tangent to the orbit; acceleration is directed toward the center of the Earth.

Question 11

A projectile is launched from a height $$h$$ above the ground with initial velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. Which equation gives the total time in the air?

Accepted Answer

$$t = \frac{v_0$ \sin 	heta + \sqrt{(v_0$ \sin \$$theta)^2$$ + 2gh}}{g}$$

Question 12

A ball is dropped from rest from a height of $$20\ \mathrm{m}. $$What is its speed just before hitting the ground? ($$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$
v^2$ = v_0$^2 + 2 g h = 0 + 2 \cdot 9.8 \cdot 20 = 392

v = \sqrt{392} \approx 19.8\ \mathrm{m/s}$$
$$
19.8$$\ \mathrm{m/s}$

Question 13

A car rounds a flat curve of radius $$100\ \mathrm{m} at a $$speed such that the centripetal acceleration is $$4.9\ \mathrm{m/s^2$}. $$What is the speed of the car?

Accepted Answer

$$
v^2$ = a_c r = 4.9 \cdot 100 = 490

v = \sqrt{490} \approx 22.1\ \mathrm{m/s}$$
$$
22.1$$\ \mathrm{m/s}$

Question 14

A ball is thrown straight up with an initial velocity of $$15\ \mathrm{m/s}. $$How long does it take to return to the thrower's hand? ($$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$
t = \frac{2 v_0$}{g} = \frac{2 \cdot 15}{9.8} \approx 3.06\ \mathrm{s}$$
$$
3.06$$\ \mathrm{s}$

Question 15

A ball is thrown horizontally from a $$20\ \mathrm{m}$$ high building with a speed of $$5\ \mathrm{m/s}. $$What is the speed of the ball just before it hits the ground?

Accepted Answer

$$
	ext{Time to fall: } t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{40}{9.8}} \approx 2.02\ \mathrm{s}$$

ext{Vertical speed: } v_y = g t = 9.8 \cdot 2.02 \approx 19.8\ \mathrm{m/s}$$

ext{Total speed: } v = \sqrt{v_x^2$ + v_y^2$} = \sqrt{5^2$ + 19.8^2$} \approx \sqrt{25 + 392} = \sqrt{417} \approx 20.4\ \mathrm{m/s}$$
$$
20.4$$\ \mathrm{m/s}$

Question 16

A car travels around a circular track of radius $$r at a $$constant speed $$v. $$What is the period $$T of $$one revolution?

Accepted Answer

$$T = \frac{2 \pi r}{v}$$

Question 17

A ball is thrown at $$45^\circ$$ above the horizontal with a speed of $$10\ \mathrm{m/s}. $$What is the time to reach maximum height? ($$g = 9.8\ \mathrm{m/s^2$}$$, $$\sin 45^\circ = \cos 45^\circ = 0.707)$$

Accepted Answer

$$
v_{y0} = 10 \cdot 0.707 = 7.07\ \mathrm{m/s}

t$$ = \frac{v_{y0}}{g} = \frac{7.07}{9.8} \approx 0.722\ \mathrm{s}$$
$$
$$0.722\ \mathrm{s}$$

Projectile Motion and Uniform Circular Motion: Study Notes for Physics with Calculus

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