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 the time the projectile spends in the air?

$$t = \dfrac{2 v_0$ \sin \theta}{g}$$

Q: A puck on an air table initially at rest receives an acceleration $$\vec{a} = 10\ (\mathrm{m/s}^2) \hat{x} + 20\ (\mathrm{m/s}^2) \hat{y}. $$What is the magnitude of its velocity after $$0.10$$ seconds?

$$v = \sqrt{(10 \times 0.10)^2$ + (20 \times 0.10)^2$} = \sqrt{1^2$ + 2^2$} = \sqrt{5} = 2.24\ \mathrm{m/s}$$

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

$$\alpha = \dfrac{33.33 \times 2\pi}{60 \times 5.00} = 0.698\ \mathrm{rad/s}^2$$

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

$$\omega = \dfrac{4.0 \times 2\pi}{5.0 \times 60} = 0.084\ \mathrm{rad/s}$$

Q: A disk rotates at $$200$$ rpm. What is its angular speed in $$\mathrm{rad/s}$$?

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

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 the time the projectile spends in the air?

Accepted Answer

$$t = \dfrac{2 v_0$ \sin 	heta}{g}$$

Question 3

A ball is thrown straight up from the edge of a cliff with initial speed $$v_0. $$Another ball is thrown straight down with the same speed from the same height. Neglecting air resistance, which ball hits the ground with greater speed?

Accepted Answer

Both balls hit the ground with the same speed

Question 4

A hockey puck starts at $$(x_0$, y_0$) = (0, 0)$$ and moves with constant velocity $$v_x = 2\ \mathrm{m/s}$$, $$v_y = 1\ \mathrm{m/s}. $$How far does it travel in 5 seconds?

Accepted Answer

$$d = \sqrt{(10)^2$ + (5)^2$} = \sqrt{100 + 25} = \sqrt{125} = 11.2\ \mathrm{m}$$

Question 5

A puck on an air table initially at rest receives an acceleration $$\vec{a} = 10\ (\mathrm{m/s}^2) \hat{x} + 20\ (\mathrm{m/s}^2) \hat{y}. $$What is the magnitude of its velocity after $$0.10$$ seconds?

Accepted Answer

$$v = \sqrt{(10 	imes 0.10)^2$ + (20 	imes 0.10)^2$} = \sqrt{1^2$ + 2^2$} = \sqrt{5} = 2.24\ \mathrm{m/s}$$

Question 6

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 maximum height reached by the projectile?

Accepted Answer

$$h = \dfrac{(100 \sin 30^\$$circ)^2$$}{2g}$$

Question 7

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

Accepted Answer

$$\alpha = \dfrac{33.33 	imes 2\pi}{60 	imes 5.00} = 0.698\ \mathrm{rad/s}^2$$

Question 8

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

Accepted Answer

$$\omega = \dfrac{4.0 	imes 2\pi}{5.0 	imes 60} = 0.084\ \mathrm{rad/s}$$

Question 9

A disk rotates at $$200$$ rpm. What is its angular speed in $$\mathrm{rad/s}$$?

Accepted Answer

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

Question 10

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 11

A coin is placed at the edge of a rotating disk and another near the center. If the disk makes one revolution, which coin travels a greater distance?

Accepted Answer

The coin at the edge

Question 12

A merry-go-round rotates at a constant angular speed. Two bugs sit at different radii. How do their tangential speeds compare?

Accepted Answer

The bug farther from the center moves faster

Question 13

A projectile is launched horizontally from a table. At the same instant, another ball is dropped from the same height. Which hits the ground first?

Accepted Answer

Both hit the ground at the same time

Question 14

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

Accepted Answer

Directly at the monkey

Question 15

What is the formula for centripetal acceleration for an object moving in a circle of radius $$r at $$speed $$v$$?

Accepted Answer

$$a_c = \dfrac{v^2$}{r}$$

Question 16

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

Accepted Answer

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

Question 17

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

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

Question 18

A boy jumps onto a merry-go-round (radius $$2.00\ \mathrm{m}) $$and circles for $$20\ \mathrm{s}$$ before coming to rest at $$1.75\ \mathrm{m}$$ from the axis. If the initial angular velocity is $$2.00\ \mathrm{rad/s}$$, what is the total angle he turns through?

Accepted Answer

$$	heta = 2.00 	imes 20 = 40\ \mathrm{rad}$$

Kinematics in Two Dimensions: Kinematics, Projectile Motion, and Angular Motion

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