Q: A hockey puck slides on a frictionless air table with an initial velocity of $$2.0\ \mathrm{m/s} at an $$angle of $$30^\circ$$ above the x-axis. What is the y-component of its velocity?

$$v_y = 2.0\ \mathrm{m/s} \times \sin(30^\circ) = 1.0\ \mathrm{m/s}$$

Q: A projectile is launched from the ground at a speed $$v_0$$ and angle $$\theta$$ above the horizontal. Ignoring air resistance, what is the formula for its maximum height?

$$h_{max} = \dfrac{v_0$^2 \sin^2$ \theta}{2g}$$

Q: A projectile is fired from a cannon at $$30^\circ$$ above the ground with an initial speed of $$100\ \mathrm{m/s}. $$What is the total time the projectile spends in the air? (Assume $$g = 9.81\ \mathrm{m/s^2$})$$

$$t = \dfrac{2 v_0$ \sin \theta}{g} = \dfrac{2 \times 100 \times \sin 30^\circ}{9.81} = 10.2\ \mathrm{s}$$

Q: A car makes $$4.0$$ laps around a circular track of radius $$50\ \mathrm{m} in$$ $$5.0$$ minutes. What is its average angular speed in rad/s?

$$\omega = \dfrac{4 \times 2\pi}{5 \times 60} = 0.084\ \mathrm{rad/s}$$

Q: A record accelerates from rest to $$33\frac{1}{3}$$ rpm in $$5.00$$ seconds. What is its angular acceleration in rad/s$$^2$$?

$$\alpha = \dfrac{33.33 \times 2\pi / 60}{5.00} = 0.70\ \mathrm{$$rad/s^2$$}$$

Q: A disk spins at $$200$$ rpm. What is its angular speed in rad/s?

$$\omega = \dfrac{200 \times 2\pi}{60} = 20.94\ \mathrm{rad/s}$$

Question 1

A hockey puck slides on a frictionless air table with an initial velocity of $$2.0\ \mathrm{m/s} at an $$angle of $$30^\circ$$ above the x-axis. What is the y-component of its velocity?

Accepted Answer

$$v_y = 2.0\ \mathrm{m/s} 	imes \sin(30^\circ) = 1.0\ \mathrm{m/s}$$

Question 2

A projectile is launched from the ground at a speed $$v_0$$ and angle $$	heta$$ above the horizontal. Ignoring air resistance, what is the formula for its maximum height?

Accepted Answer

$$h_{max} = \dfrac{v_0$^2 \sin^2$ 	heta}{2g}$$

Question 3

A ball is thrown straight up with initial velocity $$v_0. $$What is its velocity when it returns to the same height from which it was thrown (neglecting air resistance)?

Accepted Answer

$$-v_0$

Question 4

A hockey puck starts at $$(0,0)$$ and moves with constant velocity $$v_x = 2\ \mathrm{m/s}$$, $$v_y = 1\ \mathrm{m/s}. $$How far from the origin is the puck after $$5$$ seconds?

Accepted Answer

$$d = \sqrt{(10)^2$ + (5)^2$} = \sqrt{100 + 25} = 11.18\ \mathrm{m}$$

Question 5

A projectile is fired from a cannon at $$30^\circ$$ above the ground with an initial speed of $$100\ \mathrm{m/s}. $$What is the total time the projectile spends in the air? (Assume $$g = 9.81\ \mathrm{m/s^2$})$$

Accepted Answer

$$t = \dfrac{2 v_0$ \sin 	heta}{g} = \dfrac{2 	imes 100 	imes \sin 30^\circ}{9.81} = 10.2\ \mathrm{s}$$

Question 6

A ball is dropped from a height $$h$$ and another is thrown upward from the same height with speed $$v_0. $$Neglecting air resistance, which ball hits the ground first?

Accepted Answer

The ball dropped

Question 7

A monkey hangs from a tree. You aim and shoot a rock at the monkey at the same instant it lets go and falls. Where should you aim to hit the monkey?

Accepted Answer

Directly at the monkey

Question 8

A car makes $$4.0$$ laps around a circular track of radius $$50\ \mathrm{m} in$$ $$5.0$$ minutes. What is its average angular speed in rad/s?

Accepted Answer

$$\omega = \dfrac{4 	imes 2\pi}{5 	imes 60} = 0.084\ \mathrm{rad/s}$$

Question 9

A record accelerates from rest to $$33\frac{1}{3}$$ rpm in $$5.00$$ seconds. What is its angular acceleration in rad/s$$^2$$?

Accepted Answer

$$\alpha = \dfrac{33.33 	imes 2\pi / 60}{5.00} = 0.70\ \mathrm{$$rad/s^2$$}$$

Question 10

A coin sits at the edge of a rotating disk of radius $$r$$ and angular speed $$\omega. $$What is the magnitude of its centripetal acceleration?

Accepted Answer

$$a_c = r \$$omega^2$$

Question 11

A merry-go-round rotates at a constant angular speed $$\omega. $$Two children sit at radii $$r_1$$ and $$r_2$$ ($$r_2$$ \u003e $$r_1). $$How do their tangential speeds compare?

Accepted Answer

The child at $$r_2$$ moves faster

Question 12

A disk spins at $$200$$ rpm. What is its angular speed in rad/s?

Accepted Answer

$$\omega = \dfrac{200 	imes 2\pi}{60} = 20.94\ \mathrm{rad/s}$$

Question 13

A fighter jet pulls out of a dive at $$80\ \mathrm{m/s} in a $$circular path of radius $$150\ \mathrm{m}. $$What is the magnitude of the centripetal acceleration?

Accepted Answer

$$a_c = \dfrac{v^2$}{r} = \dfrac{(80)^2$}{150} = 42.7\ \mathrm{m/s^2$}$$

Question 14

A stick rotates about a fixed pole. Five people hold the stick at different distances from the pole. Which person moves fastest?

Accepted Answer

The person farthest from the pole

Question 15

A ball is launched horizontally from a table of height $$h$$ with speed $$v_0. $$How long does it take to hit the ground?

Accepted Answer

$$t = \sqrt{\dfrac{2h}{g}}$$

Question 16

A record player is stopped from spinning at $$33\frac{1}{3}$$ rpm to rest in $$0.50$$ seconds. What is the angular acceleration?

Accepted Answer

$$\alpha = -\dfrac{33.33 	imes 2\pi / 60}{0.50} = -6.98\ \mathrm{$$rad/s^2$$}$$

Question 17

A boy jumps onto a merry-go-round of radius $$2.00\ \mathrm{m}$$ and lands at $$1.75\ \mathrm{m}$$ from the center. If the angular velocity is $$2.00\ \mathrm{rad/s}$$, what is his tangential speed?

Accepted Answer

$$v = r \omega = 1.75 	imes 2.00 = 3.50\ \mathrm{m/s}$$

Question 18

A projectile is launched at $$v_0$$ and angle $$	heta. $$What is the formula for its horizontal range (assuming it lands at the same height)?

Accepted Answer

$$R = \dfrac{v_0$^2 \sin 2	heta}{g}$$

Kinematics in Two Dimensions: Kinematics, Projectile Motion, and Angular Motion (Physics 130, Lecture 4)

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