Q: A motorcycle is launched off a ramp as shown in the image. If the initial velocity of the motorcycle is $v_0$ at an angle $\theta$ above the horizontal, which of the following correctly gives the horizontal and vertical components of the initial velocity?

$v_{0x} = v_0 \cos \theta$, $v_{0y} = v_0 \sin \theta$

Q: A projectile is launched from the ground with initial speed $v_0$ at angle $\theta$. What is the time of flight until it returns to the ground?

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

Q: A fan slows down from 60 rpm to rest in 25 s. What is the angular acceleration, assuming it is constant?

$\alpha = -\frac{\omega_0}{t} = -\frac{2\pi \times 60/60}{25}$ \text{ rad/s}^2$

Q: A projectile is launched from the ground with initial speed $v_0$ at angle $\theta$. What is the maximum height reached?

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

Q: A projectile is launched from the ground with initial speed $v_0$ at angle $\theta$. What is the horizontal range?

$R = \frac{v_0^2 \sin 2\theta}{g}$

Question 1

A motorcycle is launched off a ramp as shown in the image. If the initial velocity of the motorcycle is $v_0$ at an angle $	heta$ above the horizontal, which of the following correctly gives the horizontal and vertical components of the initial velocity?

Accepted Answer

$v_{0x} = v_0 \cos 	heta$, $v_{0y} = v_0 \sin 	heta$

Question 2

Given a particle moving along a curved path in two dimensions, its position vector at time $t$ is $\vec{r}(t) = x(t) \hat{i} + y(t) \hat{j}$. What is the correct expression for its instantaneous velocity?

Accepted Answer

$\vec{v} = \frac{dx}{dt} \hat{i} + \frac{dy}{dt} \hat{j}$

Question 3

A projectile is launched with an initial speed $v_0$ at an angle $	heta$ above the horizontal. Neglecting air resistance, what is the acceleration vector during its flight?

Accepted Answer

$\vec{a} = -g \hat{j}$

Question 4

If a particle moves in two dimensions such that its $x$-component of velocity is $v_x = 3$ m/s and its $y$-component is $v_y = 4$ m/s, what is the magnitude of its velocity?

Accepted Answer

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

Question 5

A ball is thrown horizontally from a cliff with speed $v_0$. Which statement is true about its vertical velocity after 2 seconds?

Accepted Answer

It increases linearly with time: $v_y = -g t$

Question 6

A projectile is launched from the ground with initial speed $v_0$ at angle $	heta$. What is the time of flight until it returns to the ground?

Accepted Answer

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

Question 7

Which of the following best describes the direction of the instantaneous velocity vector for a particle moving along a curved path?

Accepted Answer

It is tangent to the trajectory at that point.

Question 8

A projectile is launched at an angle $	heta$ with initial speed $v_0$. What launch angle gives the maximum range on level ground?

Accepted Answer

$	heta = 45^\circ$

Question 9

A ball moves in a circle of radius $r$ at constant speed $v$. What is the magnitude of its centripetal acceleration?

Accepted Answer

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

Question 10

A Ferris wheel rotates at a constant angular velocity $\omega$. What is the tangential speed of a rider at a distance $r$ from the center?

Accepted Answer

$v = \omega r$

Question 11

A fan slows down from 60 rpm to rest in 25 s. What is the angular acceleration, assuming it is constant?

Accepted Answer

$\alpha = -\frac{\omega_0}{t} = -\frac{2\pi 	imes 60/60}{25}$ 	ext{ rad/s}^2$

Question 12

If a particle moves in a circle of radius $r$ with angular velocity $\omega$, what is the angular displacement after time $t$?

Accepted Answer

$\Delta 	heta = \omega t$

Question 13

A projectile is launched horizontally from a cliff of height $h$ with speed $v_0$. What is the time to hit the ground?

Accepted Answer

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

Question 14

A ball is moving in a circle and its angular velocity is increasing. What is the direction of its angular acceleration?

Accepted Answer

In the same direction as the angular velocity.

Question 15

A projectile is launched from the ground with initial speed $v_0$ at angle $	heta$. What is the maximum height reached?

Accepted Answer

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

Question 16

A ball is moving in a circle of radius $r$ at constant speed $v$. What is the direction of the acceleration vector?

Accepted Answer

Toward the center of the circle.

Question 17

A person on a bicycle moves at 5 m/s east, while a car moves at 15 m/s east. What is the velocity of the car relative to the cyclist?

Accepted Answer

$v_{car,rel} = 10$ m/s east

Question 18

A projectile is launched from the ground with initial speed $v_0$ at angle $	heta$. What is the horizontal range?

Accepted Answer

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

Question 19

A ball is thrown at an angle $	heta$ above the horizontal. At the highest point of its trajectory, which statement is true?

Accepted Answer

The vertical component of velocity is zero.

Kinematics in Two Dimensions: Position, Displacement, and Velocity

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