Q: A baseball is thrown horizontally from a height of 2 meters above the ground with an initial speed of 10 m/s. Neglecting air resistance, how long does it take for the baseball to hit the ground?

$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 2}{9.8}} \approx 0.64\ \text{s}$

Q: A projectile is launched with an initial speed $v_0$ at an angle $\theta$ above the horizontal. What is the formula for its maximum height (neglecting air resistance)?

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

Q: A train moves east at 20 m/s. A passenger walks west inside the train at 2 m/s relative to the train. What is the passenger's velocity relative to the ground?

$v_{pg} = v_{pt} + v_{tg} = -2\ \text{m/s} + 20\ \text{m/s} = 18\ \text{m/s}$\ east

Q: A soccer ball is kicked with an initial velocity of $20\ \text{m/s}$ at an angle of $30^\circ$ above the horizontal. What is the initial vertical component of velocity?

$v_{0y} = v_0 \sin \theta = 20 \sin 30^\circ = 10\ \text{m/s}$

Question 1

A baseball is thrown horizontally from a height of 2 meters above the ground with an initial speed of 10 m/s. Neglecting air resistance, how long does it take for the baseball to hit the ground?

Accepted Answer

$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 	imes 2}{9.8}} \approx 0.64\ 	ext{s}$

Question 2

Which of the following statements about the velocity and acceleration vectors for a particle moving in a circle at constant speed is correct?

Accepted Answer

The velocity and acceleration vectors are always perpendicular to each other.

Question 3

A projectile is launched with an initial speed $v_0$ at an angle $	heta$ above the horizontal. What is the formula for its maximum height (neglecting air resistance)?

Accepted Answer

$H = \frac{v_0^2 \sin^2 	heta}{2g}$

Question 4

A car moves along a curved path with increasing speed. At a certain instant, which direction does the acceleration vector point?

Accepted Answer

Ahead of the normal to the path, combining tangential and normal components.

Question 5

Which equation correctly expresses the position vector $\vec{r}$ of a particle in three dimensions?

Accepted Answer

$\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$

Question 6

A train moves east at 20 m/s. A passenger walks west inside the train at 2 m/s relative to the train. What is the passenger's velocity relative to the ground?

Accepted Answer

$v_{pg} = v_{pt} + v_{tg} = -2\ 	ext{m/s} + 20\ 	ext{m/s} = 18\ 	ext{m/s}$\ east

Question 7

For a projectile launched horizontally from a cliff, which statement is true about its horizontal and vertical motions (neglecting air resistance)?

Accepted Answer

Horizontal motion has constant velocity; vertical motion has constant acceleration.

Question 8

A roller coaster car moves along a curved track and slows down. At a certain point, how does the direction of its acceleration vector compare to the normal of the path?

Accepted Answer

The acceleration vector points behind the normal to the path.

Question 9

What is the magnitude of the centripetal acceleration for an object moving in a circle of radius $R$ at speed $v$?

Accepted Answer

$a_c = \frac{v^2}{R}$

Question 10

A soccer ball is kicked with an initial velocity of $20\ 	ext{m/s}$ at an angle of $30^\circ$ above the horizontal. What is the initial vertical component of velocity?

Accepted Answer

$v_{0y} = v_0 \sin 	heta = 20 \sin 30^\circ = 10\ 	ext{m/s}$

Question 11

Which of the following best describes the acceleration vector for a particle moving along a curved path at constant speed?

Accepted Answer

The acceleration is perpendicular to the velocity and points toward the center of curvature.

Question 12

A projectile reaches its peak height. What is true about its velocity and acceleration vectors at that instant?

Accepted Answer

Velocity is horizontal; acceleration is vertical downward.

Question 13

If air resistance is significant for a baseball thrown at high speed, which of the following is true about its trajectory?

Accepted Answer

The trajectory is no longer a parabola and the range is reduced.

Question 14

A cyclist moves north at 5 m/s. A wind blows east at 3 m/s. What is the magnitude of the cyclist's velocity relative to the ground?

Accepted Answer

$v = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \approx 5.83\ 	ext{m/s}$

Question 15

Which formula gives the range $R$ of a projectile launched from ground level with initial speed $v_0$ at angle $	heta$ (neglecting air resistance)?

Accepted Answer

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

Question 16

A particle's position vector is given by $\vec{r}(t) = (2t)\hat{i} + (3t^2)\hat{j}$. What is its instantaneous velocity at $t = 1$ s?

Accepted Answer

$\vec{v}(t) = \frac{d\vec{r}}{dt} = 2\hat{i} + 6t\hat{j}$;\ at\ t=1:\ \vec{v} = 2\hat{i} + 6\hat{j}$

Question 17

A car moves in a circle of radius $R$ at a speed that increases at a constant rate $a_t$. What is the total acceleration at an instant when its speed is $v$?

Accepted Answer

$a = \sqrt{a_t^2 + \left(\frac{v^2}{R}\right)^2}$

Question 18

Which of the following is true about the components of acceleration for a particle moving in two dimensions?

Accepted Answer

Each component equals the instantaneous rate of change of the corresponding velocity component.

Chapter 03

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