Q: A car drives up a 10° slope at a constant speed of 15 m/s for 10 seconds. What is the vertical height gained by the car during this time?

$$\Delta y = 15 \ \mathrm{m/s} \times 10 \ \mathrm{s} \times \sin 10^\circ = 26 \ \mathrm{m}$$

Q: A skier starts from rest and accelerates down a frictionless slope of length 360 m and vertical drop 170 m. What is the skier's speed at the bottom of the slope?

$$v = \sqrt{2 a_x \Delta x} = \sqrt{2 \times 4.6 \ \mathrm{m/s}^2 \times 360 \ \mathrm{m}} = 58 \ \mathrm{m/s}$$

Q: A projectile is launched from the ground with initial speed $$v_0 at an $$angle $$\theta$$ above the horizontal. What is the time of flight until it lands back on the ground (neglecting air resistance)?

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

Q: A dog runs horizontally off a dock 0.61 m above the water at 8.5 m/s. How far from the dock does the dog land?

$$\Delta x = v_{0x} t = 8.5 \ \mathrm{m/s} \times \sqrt{\frac{2 \times 0.61 \ \mathrm{m}}{9.8 \ \mathrm{m/s}^2}} = 3.0 \ \mathrm{m}$$

Q: A vector $$\vec{A}$$ has components $$A_x = 3$$ and $$A_y = 4. $$What is the magnitude and direction of $$\vec{A}$$?

$$A = \sqrt{3^2$ + 4^2$} = 5$$, $$\theta = \tan$$^{-1}$$(4/3) = 53.1^\circ$$

Q: A vector $$\vec{A}$$ has magnitude $$A$$ and makes an angle $$\theta$$ with the x-axis. What are its components?

$$A_x = A \cos \theta$$, $$A_y = A \sin \theta$$

Question 1

A car drives up a 10° slope at a constant speed of 15 m/s for 10 seconds. What is the vertical height gained by the car during this time?

Accepted Answer

$$\Delta y = 15 \ \mathrm{m/s} 	imes 10 \ \mathrm{s} 	imes \sin 10^\circ = 26 \ \mathrm{m}$$

Question 2

A skier starts from rest and accelerates down a frictionless slope of length 360 m and vertical drop 170 m. What is the skier's speed at the bottom of the slope?

Accepted Answer

$$v = \sqrt{2 a_x \Delta x} = \sqrt{2 	imes 4.6 \ \mathrm{m/s}^2 	imes 360 \ \mathrm{m}} = 58 \ \mathrm{m/s}$$

Question 3

A projectile is launched from the ground with initial speed $$v_0 at an $$angle $$	heta$$ above the horizontal. What is the time of flight until it lands back on the ground (neglecting air resistance)?

Accepted Answer

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

Question 4

A dog runs horizontally off a dock 0.61 m above the water at 8.5 m/s. How far from the dock does the dog land?

Accepted Answer

$$\Delta x = v_{0x} t = 8.5 \ \mathrm{m/s} 	imes \sqrt{\frac{2 	imes 0.61 \ \mathrm{m}}{9.8 \ \mathrm{m/s}^2}} = 3.0 \ \mathrm{m}$$

Question 5

Which of the following statements about projectile motion is correct (neglecting air resistance)?

Accepted Answer

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

Question 6

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

Accepted Answer

$$A = \sqrt{3^2$ + 4^2$} = 5$$, $$	heta = 	an$$^{-1}$$(4/3) = 53.1^\circ$$

Question 7

A crate slides down a frictionless ramp inclined at angle $$	heta. $$What is the acceleration of the crate along the ramp?

Accepted Answer

$$a = g \sin 	heta$$

Question 8

A hiker walks 4 miles east and then 3 miles north. What is the magnitude of the net displacement?

Accepted Answer

$$\sqrt{4^2$ + 3^2$} = 5$$ miles

Question 9

If a vector $$\vec{A} is $$multiplied by $$-1$$, what happens to the vector?

Accepted Answer

Its direction is reversed but its magnitude remains the same.

Question 10

A ball is thrown horizontally from a cliff with speed $$v_0. $$Which equation gives the vertical position $$y as a $$function of time $$t ($$take $$y=0 at $$the launch point, downward positive)?

Accepted Answer

$$y = \frac{1}{2} g t^2$

Question 11

A velocity-time graph for an object moving with constant acceleration is a straight line. What does the area under this line represent?

Accepted Answer

The object's displacement during the time interval.

Question 12

A projectile is launched at an angle $$	heta$$ with initial speed $$v_0. $$For which angle is the range maximized (neglecting air resistance)?

Accepted Answer

$$45^\circ$$

Question 13

A vector $$\vec{A}$$ has magnitude $$A$$ and makes an angle $$	heta$$ with the x-axis. What are its components?

Accepted Answer

$$A_x = A \cos 	heta$$, $$A_y = A \sin 	heta$$

Question 14

A ball is dropped from rest and another is thrown horizontally from the same height at the same instant. Which hits the ground first?

Accepted Answer

They hit at the same time.

Question 15

A particle moves in two dimensions with velocity components $$v_x = 3 \ \mathrm{m/s}$$ and $$v_y = 4 \ \mathrm{m/s}. $$What is the magnitude of its velocity?

Accepted Answer

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

Question 16

A car rounds a curve at constant speed. What can you say about its acceleration?

Accepted Answer

The car is accelerating because its velocity direction is changing.

Vectors and Motion in Two Dimensions: Physics with Calculus Study Notes

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