Q: A boat moves with velocity $$\textbf{v}_{BW}$$ relative to the water, and the water moves with velocity $$\textbf{v}_{WS}$$ relative to the shore. What is the velocity of the boat relative to the shore?

$$\textbf{v}_{BS} = \textbf{v}_{BW} + \textbf{v}_{WS}$$

Q: A vector $$\textbf{C} is $$subtracted from vector $$\textbf{D}$$, written as $$\textbf{D} - \textbf{C}. $$Which statement is correct?

Subtracting $$\textbf{C} is $$equivalent to adding $$-\textbf{C} to$$ $$\textbf{D}.$$

Q: A projectile is launched from the ground with initial velocity $$v_0 at $$angle $$\theta. $$What is the equation for its trajectory $$y as a $$function of $$x$$?

$$y = x \tan \theta - \frac{g x^2$}{2 v_0$^2 \cos^2$ \theta}$$

Q: A rescue plane flying horizontally at $$v = 60\,\text{m/s}$$ drops a package from a height of $$h = 80\,\text{m}. $$Neglecting air resistance, how far horizontally from the release point will the package land? ($$g = 9.8\,\text{m/s}^2)$$

$$x = v t = 60 \times \sqrt{\frac{2 \times 80}{9.8}} \approx 60 \times 4.04 = 242\,\text{m}$$

Q: A vector $$\textbf{E}$$ has components $$E_x = -6$$ and $$E_y = 8. $$What is the direction of $$\textbf{E}$$ relative to the positive x-axis?

$$\theta = \arctan \left( \frac{E_y}{E_x} \right ) = \arctan \left( \frac{8}{-6} \right ) \approx -53.1^\text{o} ($$measured clockwise from x-axis)

Question 1

Which of the following quantities is a vector?

Accepted Answer

Momentum

Question 2

A vector $$	extbf{A}$$ has components $$A_x = 3$$ and $$A_y = 4. $$What is the magnitude of $$	extbf{A}$$?

Accepted Answer

$$5$$

Question 3

If a vector $$	extbf{V} is $$multiplied by a scalar $$-2$$, what happens to the direction and magnitude of the resulting vector?

Accepted Answer

The magnitude doubles and the direction reverses.

Question 4

A projectile is launched from ground level with an initial velocity $$v_0 at an $$angle $$	heta$$ above the horizontal. Which equation gives the horizontal component of the initial velocity?

Accepted Answer

$$v_{0x} = v_0$ \cos 	heta$$

Question 5

A ball is thrown horizontally from a cliff. Which statement best describes its motion?

Accepted Answer

The horizontal velocity remains constant while the vertical velocity increases due to gravity.

Question 6

A vector $$	extbf{B}$$ has a magnitude of $$10$$ and points at $$30^	ext{o}$$ above the x-axis. What is the y-component of $$	extbf{B}$$?

Accepted Answer

$$B_y = 10 \sin 30^	ext{o} = 5$$

Question 7

Which method is NOT used for graphical vector addition?

Accepted Answer

Component method

Question 8

A projectile is launched at $$v_0 = 20$$\,	ext{m/s}$$ at an angle 45^	$$ext{o}$$ above the horizontal. What is the maximum height reached by the projectile? (Assume g = 9.8$$\,	ext{m/s}^2$$)

Accepted Answer

$$H = \frac{(v_0$ \sin \$$theta)^2$$}{2g} = \frac{(20 \sin 45^	ext{o})^2$}{2 	imes 9.8} \approx 10.2\,	ext{m}$$

Question 9

A boat moves with velocity $$	extbf{v}_{BW}$$ relative to the water, and the water moves with velocity $$	extbf{v}_{WS}$$ relative to the shore. What is the velocity of the boat relative to the shore?

Accepted Answer

$$	extbf{v}_{BS} = 	extbf{v}_{BW} + 	extbf{v}_{WS}$$

Question 10

A vector $$	extbf{C} is $$subtracted from vector $$	extbf{D}$$, written as $$	extbf{D} - 	extbf{C}. $$Which statement is correct?

Accepted Answer

Subtracting $$	extbf{C} is $$equivalent to adding $$-	extbf{C} to$$ $$	extbf{D}.$$

Question 11

A projectile is launched from the ground with initial velocity $$v_0 at $$angle $$	heta. $$What is the equation for its trajectory $$y as a $$function of $$x$$?

Accepted Answer

$$y = x 	an 	heta - \frac{g x^2$}{2 v_0$^2 \cos^2$ 	heta}$$

Question 12

Which of the following is NOT a scalar quantity?

Accepted Answer

Force

Question 13

A rescue plane flying horizontally at $$v = 60\,	ext{m/s}$$ drops a package from a height of $$h = 80\,	ext{m}. $$Neglecting air resistance, how far horizontally from the release point will the package land? ($$g = 9.8\,	ext{m/s}^2)$$

Accepted Answer

$$x = v t = 60 	imes \sqrt{\frac{2 	imes 80}{9.8}} \approx 60 	imes 4.04 = 242\,	ext{m}$$

Question 14

Which step is NOT part of the recommended procedure for solving projectile motion problems?

Accepted Answer

Combine x and y motions into a single equation before solving.

Question 15

A vector $$	extbf{E}$$ has components $$E_x = -6$$ and $$E_y = 8. $$What is the direction of $$	extbf{E}$$ relative to the positive x-axis?

Accepted Answer

$$	heta = \arctan \left( \frac{E_y}{E_x} ight ) = \arctan \left( \frac{8}{-6} ight ) \approx -53.1^	ext{o} ($$measured clockwise from x-axis)

Question 16

A car moves east at $$20\,	ext{m/s}$$ relative to the ground, while a passenger walks north at $$2\,	ext{m/s}$$ relative to the car. What is the passenger's velocity relative to the ground?

Accepted Answer

$$\sqrt{20^2$ + 2^2$} = \sqrt{404} \approx 20.1\,	ext{m/s} at$$ $$\arctan(2/20) \approx 5.7^	ext{o}$$ north of east

Question 17

Which statement about projectile motion is correct?

Accepted Answer

The horizontal and vertical motions are independent except for sharing the same time interval.

Question 18

A vector $$	extbf{F}$$ has components $$F_x = 7$$ and $$F_y = 24. $$What is the magnitude of $$	extbf{F}$$?

Accepted Answer

$$\sqrt{7^2$ + 24^2$} = \sqrt{49 + 576} = \sqrt{625} = 25$$

Kinematics in Two Dimensions and Vectors: Study Notes

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