Q: A car accelerates from rest along a straight ramp of length $$120\ \mathrm{m}$$ and reaches a speed of $$20\ \mathrm{m/s} at $$the end. What is the car's acceleration?

$$a = \frac{v^2$}{2d} = \frac{(20)^2$}{2 \times 120} = 1.67\ \mathrm{m/s^2$}$$

Q: A particle's position is given by $$x = 2.0\ \mathrm{m} - (0.25\ \mathrm{m/s^2$})t^2$ and y = (1.0$$\ \mathrm{m/s})t + (0.025\ \mathrm{$$m/s^3$$}$$)t^3. $$What is the $$y-$$component of the instantaneous velocity at $$t = 2.0\ \mathrm{s}$$?

$$v_y = \frac{dy}{dt} = 1.0 + 3 \times 0.025 \times (2.0)^2$ = 1.3\ \mathrm{m/s}$$

Question 1

A projectile is launched from the ground with an initial speed $$v_0 at an $$angle $$\$$alpha_0$$ above the horizontal. Neglecting air resistance, which of the following correctly describes the horizontal and vertical components of the projectile's acceleration during its flight?

Accepted Answer

$$a_x = 0,\ a_y = -g$$

Question 2

A car moves at constant speed around a circular track of radius $$R. $$Which of the following statements about its acceleration is correct?

Accepted Answer

The acceleration is always directed toward the center of the circle.

Question 3

A ball is thrown horizontally from a cliff with speed $$v_0. At $$the same instant, another ball is dropped from the same height. Neglecting air resistance, which statement is true about the time it takes for both balls to hit the ground?

Accepted Answer

Both balls hit the ground at the same time.

Question 4

The position vector of a particle is given by $$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}. $$Which of the following expressions gives the instantaneous velocity vector?

Accepted Answer

$$\vec{v} = \frac{d\vec{r}}{dt}$$

Question 5

A roller coaster car moves along a curved track at increasing speed. At a certain point, which of the following best describes the direction of its acceleration vector?

Accepted Answer

It has a component tangent to the path and a component toward the center of curvature.

Question 6

A baseball is hit with an initial velocity of $$50\ \mathrm{m/s} at an $$angle of $$53.1^\circ$$ above the horizontal. If air resistance is not negligible, what happens to the range and maximum height compared to the case with no air resistance?

Accepted Answer

Both the range and maximum height decrease.

Question 7

A particle moves in the $$xy-$$plane 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

$$5\ \mathrm{m/s}$$

Question 8

A car accelerates from rest along a straight ramp of length $$120\ \mathrm{m}$$ and reaches a speed of $$20\ \mathrm{m/s} at $$the end. What is the car's acceleration?

Accepted Answer

$$a = \frac{v^2$}{2d} = \frac{(20)^2$}{2 	imes 120} = 1.67\ \mathrm{m/s^2$}$$

Question 9

Which of the following best describes the effect of air resistance on the trajectory of a projectile?

Accepted Answer

The trajectory is no longer a parabola, and both range and maximum height decrease.

Question 10

A boat moves east across a river at $$5\ \mathrm{m/s}$$ relative to the water. The river flows north at $$3\ \mathrm{m/s}. $$What is the magnitude of the boat's velocity relative to the ground?

Accepted Answer

$$\sqrt{5^2$ + 3^2$} = 5.83\ \mathrm{m/s}$$

Question 11

A particle's position is given by $$x = 2.0\ \mathrm{m} - (0.25\ \mathrm{m/s^2$})t^2$ and y = (1.0$$\ \mathrm{m/s})t + (0.025\ \mathrm{$$m/s^3$$}$$)t^3. $$What is the $$y-$$component of the instantaneous velocity at $$t = 2.0\ \mathrm{s}$$?

Accepted Answer

$$v_y = \frac{dy}{dt} = 1.0 + 3 	imes 0.025 	imes (2.0)^2$ = 1.3\ \mathrm{m/s}$$

Question 12

Which of the following is the correct formula for the magnitude of the centripetal (radial) acceleration for an object in uniform circular motion?

Accepted Answer

$$a_{rad} = \frac{v^2$}{R}$$

Question 13

A projectile is launched with initial velocity components $$v_{0x}$$ and $$v_{0y}. $$Which equation gives the $$y-$$coordinate as a function of time, assuming $$y_0 = 0$ and neglecting air resistance?

Accepted Answer

$$y = v_{0y} t - \frac{1}{2} g t^2$

Question 14

A car is moving in a circle and begins to slow down. Which of the following best describes the direction of its acceleration?

Accepted Answer

It has a component toward the center and a component opposite to the direction of motion.

Question 15

Which of the following best describes the velocity of a particle as seen from two different frames of reference A and B?

Accepted Answer

$$\vec{v}_{P/A} = \vec{v}_{P/B} + \vec{v}_{B/A}$$

Question 16

A ball is thrown at an angle $$\$$alpha_0$$ with initial speed v_0$. What is the horizontal range of the projectile if it lands at the same height from which it was launched (neglecting air resistance)?

Accepted Answer

$$R = \frac{v_0$^2 \sin 2\$$alpha_0$$}{g}$$

Question 17

A particle moves in a circle of radius $$R$$ with varying speed. Which of the following statements is true?

Accepted Answer

The acceleration has both radial and tangential components.

Motion in Two or Three Dimensions: Study Notes

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