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?

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

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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