Q: A projectile is launched with an initial velocity $$v_0 at an $$angle $$\theta$$ above the horizontal. What is the formula for the horizontal range $$R of $$the projectile (assuming it lands at the same height it was launched)?

$$R = \dfrac{v_0$^2 \sin(2\theta)}{g}$$

Q: A baseball is hit horizontally from a height of $$1.5\ \mathrm{m}$$ above the ground with a velocity of $$20\ \mathrm{m/s}. $$How long does it take to hit the ground? (Neglect air resistance, $$g = 9.8\ \mathrm{m/s^2$})$$

$$t = \sqrt{\dfrac{2 \times 1.5}{9.8}}$$

Question 1

A soccer ball is kicked from the ground with an initial velocity of $$v_0 at an $$angle $$	heta$$ above the horizontal. Neglecting air resistance, which of the following correctly describes the acceleration of the ball during its flight?

Accepted Answer

The acceleration is $$a_x = 0$$ and $$a_y = -g at $$all times.

Question 2

A projectile is launched with an initial velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. What is the formula for the horizontal range $$R of $$the projectile (assuming it lands at the same height it was launched)?

Accepted Answer

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

Question 3

A baseball is hit horizontally from a height of $$1.5\ \mathrm{m}$$ above the ground with a velocity of $$20\ \mathrm{m/s}. $$How long does it take to hit the ground? (Neglect air resistance, $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$t = \sqrt{\dfrac{2 	imes 1.5}{9.8}}$$

Question 4

Which of the following statements about the instantaneous velocity vector of a particle moving in a plane is correct?

Accepted Answer

It is always tangent to the path of the particle.

Question 5

A car moves from point $$P_1 at$$ $$(4.0\ \mathrm{m}, 2.0\ \mathrm{m}) to $$point $$P_2 at$$ $$(7.0\ \mathrm{m}, 6.0\ \mathrm{m}) in$$ $$0.5\ \mathrm{s}. $$What is the magnitude of its average velocity?

Accepted Answer

$$v_{av} = \dfrac{\sqrt{(7.0-4.0)^2$ + (6.0-2.0)^2$}}{0.5}$$

Question 6

A projectile is launched with initial velocity components $$v_{0x}$$ and $$v_{0y}. $$Which equation gives its position in the $$y-$$direction as a function of time $$t$$?

Accepted Answer

$$y(t) = y_0$ + v_{0y} t - \dfrac{1}{2} g t^2$

Question 7

Which of the following best describes the average acceleration vector of a particle moving in a plane?

Accepted Answer

It is given by $$\vec{a}_{av} = \dfrac{\Delta \vec{v}}{\Delta t}.$$

Question 8

A paintball is fired horizontally from a gun at $$v_0 = 50$$\ \mathrm{m/s}$$ from a height of 1.5$$\ \mathrm{m}$$. How far does it travel horizontally before hitting the ground? (g = 9.8$$\ \mathrm{$$m/s^2$$}$$)

Accepted Answer

$$d = v_0$ \sqrt{\dfrac{2 h}{g}}$$

Question 9

A projectile is launched at an angle $$	heta$$ with initial speed $$v_0. $$What is the initial vertical component of velocity?

Accepted Answer

$$v_{0y} = v_0$ \sin 	heta$$

Question 10

A ball is thrown with velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. At the peak of its trajectory, which statement is true?

Accepted Answer

The vertical component of velocity is zero.

Question 11

Which of the following is true about the addition of two vectors $$\vec{A}$$ and $$\vec{B}$$ using the head-to-tail method?

Accepted Answer

The resultant vector $$\vec{A} + \vec{B} is $$drawn from the tail of $$\vec{A} to $$the tip of $$\vec{B}.$$

Question 12

A runner moves with velocity $$\vec{v}_1$$ relative to the ground. If the ground itself is moving with velocity $$\vec{v}_2$$ relative to an observer, what is the runner's velocity relative to the observer?

Accepted Answer

$$\vec{v}_{runner, observer} = \vec{v}_1 + \vec{v}_2$$

Question 13

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

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

Question 14

Which of the following is the correct expression for the magnitude of a position vector $$\vec{r} in $$Cartesian coordinates $$(x, y)$$?

Accepted Answer

$$|\vec{r}| = \sqrt{x^2$ + y^2$}$$

Question 15

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

Accepted Answer

$$v_{0x} = 25 \cos 30^\circ$$

Question 16

Which of the following is true about the vertical motion of a projectile under gravity?

Accepted Answer

It is identical to the motion of an object in free fall.

Question 17

A ball is thrown horizontally from a cliff. Which statement best describes its velocity just before hitting the ground?

Accepted Answer

It has both horizontal and vertical components.

Question 18

A projectile is launched at $$45^\circ$$ with initial speed $$v_0. $$What is the ratio of its maximum height $$H to $$its range $$R$$?

Accepted Answer

$$\dfrac{H}{R} = \dfrac{1}{4}$$

Motion in a Plane: Position, Velocity, and Acceleration in Two Dimensions

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