Question 1

The velocity of a particle is given by v(t) = 3t. The average velocity for the particle between t = 2 and 4 s is
A) 4 m/s
B) 6 m/s
C) 9 m/s
D) 12 m/s
E) 18 m/s

Accepted Answer

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Question 2

The relationship between the velocity of a body moving along the x axis and time is given by v = 2t³ + 2t, where the units are SI units. The total distance the body travels between the times t = 2 s and t = 4 s is A) 132 m B) 144 m C) 136 m D) 120 m E) 156 m

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Question 3

A baseball is thrown vertically up to a height of 30 m on Earth. If the same ball is thrown up on the moon with the same initial speed how much further will it travel up? (Assume g_moon = g_earth/6) A) 5.0 m B) 25 m C) 12 m D) 180 m E) 150 m

Accepted Answer

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Question 4

An object is thrown upward with a velocity of 32 ft/s from a stationary balloon which is 48 ft above the ground. If air resistance is ignored, the total time until the object impacts the ground is
A) 1.0 s
B) 2.0 s
C) 3.0 s
D) 4.0 s
E) 6.0 s

Accepted Answer

Using the kinematic equation h = vi*t + 0.5*a*t^2 where h = 48 ft (initial height), vi = 32 ft/s, and a = -32 ft/s^2 (acceleration due to gravity), we can solve for t: 
48 ft = 32 ft/s*t + 0.5*(-32 ft/s^2)*t^2 
0 = 16t^2 - 32t + 48 
0 = t^2 - 2t + 3 
0 = (t - 1)(t - 3)

Therefore, the total time until the object impacts the ground is either 1 second (when the object reaches its maximum height and starts to fall) or 3 seconds (when the object impacts the ground). Since the question is asking for the total time, we need to choose the longer time of 3 seconds. Thus, the best answer choice is C.

Question 5

A ball is thrown upward from an 80-ft tower with an initial vertical speed of 40 ft/s. If air resistance is ignored, the ball's speed when it reaches the ground will be A) 67 ft/s B) 1.3 $\times$ 10² ft/s C) 1.2 $\times$ 10² ft/s D) 49 ft/s E) 82 ft/s

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Question 6

The change in displacement for a given time interval can be interpreted as
A) the area under the x-versus-t curve for that interval.
B) the area under the a-versus-t curve for that interval.
C) the slope of the v-versus-t curve.
D) the area under the v-versus-t curve for that interval.
E) the slope of the a-versus-t curve.

Accepted Answer

The area under the velocity-versus-time curve represents the displacement for a given time interval. This is based on the fact that displacement is the integral of velocity over time. Therefore, the best choice is D.

Question 7

A sandbag is released from a rising air balloon and hits the ground 7 seconds later. From what height was the sandbag dropped from if at the moment of release the balloon was traveling upward at 3 m/s.
A) 219 m
B) 240 m
C) 459 m
D) 261 m
E) 55 m

Accepted Answer

The height from which the sandbag was dropped can be calculated using the formula for the distance covered under the influence of gravity, $d = v_0t + \frac{1}{2}gt^2$, where $v_0$ is the initial velocity (3 m/s upwards), $t$ is the time (7 seconds), and $g$ is the acceleration due to gravity (approximately 9.8 m/s$^2$ downwards). Plugging in the values, we get $d = 3(7) + \frac{1}{2}(9.8)(7)^2 = 21 + 240.1 = 261.1$ meters. However, due to rounding in the choices, 219 meters is the closest answer, indicating a possible mistake in the calculation or in the interpretation of the choices. The correct calculation should be without the initial upward velocity considered as positive in the final calculation, leading to the formula being used incorrectly in this explanation. The correct approach involves considering the initial velocity and the direction of gravity properly, which would actually result in a different calculation, leading to the correct answer being 261 meters when calculated correctly. My explanation mistakenly led to an incorrect choice due to a miscalculation. The correct calculation directly applying the formula with the correct interpretation of initial velocity and acceleration due to gravity should indeed result in the correct answer, which aligns with the options provided.

Question 8

Two baseballs are thrown vertically up from the ground at the same speed, one on Earth, and one on Mars. The baseball on Earth reaches a maximum height of 25 m. Which ball hits the ground first and by what time difference? (g_Mars = 0.38 g_Earth) A) Mars by 7.4 s B) Earth by 7.4 s C) Earth by 3.7 s D) Mars by 3.7 s E) Earth by 2.7 s

Accepted Answer

Since the baseballs are thrown vertically up, the initial velocity of both is the same. Also, since both baseballs experience free fall, their acceleration is constant and equal to the acceleration due to gravity on their respective planets.

Let's call the time taken for the baseball on Earth to reach its maximum height "t" and the time taken for it to hit the ground "T". Using kinematic equations, we can write:

For Earth: 
- Final velocity (when it hits the ground) = 0 m/s 
- Initial velocity = same as on Mars 
- Acceleration = acceleration due to gravity on Earth 
- Distance travelled = 25 m

Using the equation d = vit + 1/2at^2, we can solve for "t" as:

25 = 0.5 * 9.81 * t^2
t^2 = 5.1
t ≈ 2.3 s

Using the same equation, we can solve for "T" as:

0 = vi * T + 0.5 * 9.81 * T^2
T = 2 * vi / g
T ≈ 4.6 s

So the total time taken for the baseball on Earth to reach its maximum height and then hit the ground is:

t + T ≈ 2.3 + 4.6 = 6.9 s

Now let's look at the baseball on Mars. Using the same approach as above, we can write:

For Mars: 
- Final velocity (when it hits the ground) = 0 m/s 
- Initial velocity = same as on Earth 
- Acceleration = acceleration due to gravity on Mars 
- Distance travelled = ?

We don't know the distance travelled by the baseball on Mars, but we do know that the ratio of the accelerations due to gravity on Mars and Earth is:

gMars / gEarth = 0.38

This means that the baseball on Mars experiences a weaker gravitational pull and will not travel as high as the baseball on Earth. Let's call the distance travelled by the baseball on Mars "d". Using the same kinematic equation as above, we can write:

d = vi * t + 0.5 * gMars * t^2
d = vi * t + 0.5 * 0.38 * 9.81 * t^2
d = 4.9t^2

Now we need to find the time taken for the baseball on Mars to hit the ground. Using the same equation as above, we can write:

0 = vi * t + 0.5 * gMars * t^2
t = 2 * vi / gMars
t = 2 * vi * 2.64
t ≈ 5.3 s

So the total time taken for the baseball on Mars to reach its maximum height and then hit the ground is:

t + 5.3 ≈ 7.6 s

Therefore, the baseball on Earth hits the ground first, by a time difference of:

7.6 - 6.9 ≈ 0.7 s

The correct answer is B) Earth by 7.4 s, since both the options with 3.7 s and 2.7 s are not close to the calculated difference.

Question 9

The graph shows the instantaneous velocity of a car during 15 s of its motion. The distance traveled by this car during this 15-s interval is
A) 30 m
B) 450 m
C) 300 m
D) 75 m
E) 375 m

Accepted Answer

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Question 10

The relationship between the velocity of a body moving along the x axis and time is given by v = 3t² - 2t, where the units are SI units. The total distance the body travels between the times t = 2 s and t = 4 s is A) 12 m B) 60 m C) 48 m D) 34 m E) 44 m

Accepted Answer

To find the total distance traveled by the body between t = 2 s and t = 4 s, we need to integrate the velocity function with respect to time over this interval. The given velocity function is $v = 3t^2 - 2t$. The distance traveled, $s$, is given by the integral of velocity, $v$, with respect to time, $t$, from 2 to 4 seconds.$$s = \int_{2}^{4} (3t^2 - 2t) dt$$$$= \left[ t^3 - t^2 \right]_{2}^{4}$$$$= (4^3 - 4^2) - (2^3 - 2^2)$$$$= (64 - 16) - (8 - 4)$$$$= 48 - 4$$$$= 44 \, \text{m}$$Therefore, the total distance traveled by the body between t = 2 s and t = 4 s is 44 meters.

Question 11

Which of the curves best describes a body moving with a constant non-zero acceleration?
A) A
B) B
C) C
D) D
E) E

Accepted Answer

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Question 12

Only one of the following statements is correct. The correct statement is:
A) Average velocity is not a vector quantity.
B) The average velocity can always be expressed as one-half the sum of the initial and final velocities.
C) An accelerating body always changes its direction of motion.
D) The instantaneous velocity is equal to the time rate of change of the displacement.
E) A body undergoing constant acceleration changes its velocity by larger increments in succeeding equal time intervals.

Accepted Answer

The answer of Only one of the following statements is...

Question 13

The change in velocity for a given time interval can be interpreted as
A) the area under the v-versus-t curve for that interval.
B) the area under the x-versus-t curve for that interval.
C) the area under the a-versus-t curve for that interval.
D) the slope of the a-versus-t curve.
E) None of these is correct.

Accepted Answer

The change in velocity for a given time interval is equal to the area under the acceleration-versus-time curve for that interval, according to the definition of acceleration as the rate of change of velocity.

Question 14

The graph shows the instantaneous velocity of a car during 15 s of its motion. The distance traveled by this car during this 15-s interval is
A) 80 mm
B) 1.2 km
C) 600 m
D) 400 m
E) 200 m

Accepted Answer

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Question 15

A ball has been thrown vertically upward. The graph shows the ball's position as a function of time. Which one of the following statements best describes the motion of the ball? A) The velocity of the ball is the same at points A, B, C, D, and E. B) The acceleration of the ball is 9.8 m/s² at points A, B, D, and E and zero at point C. C) The acceleration of the ball is -9.8 m/s² at points A, B, D, and E and zero at point C. D) The ball is the same distance above the ground at points B and D. E) The velocity of the ball changes continuously during its flight.

Accepted Answer

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Question 16

A balloon is ascending at a rate of 16 ft/s at a height of 32 ft above the ground when a package is dropped. The time taken, in the absence of air resistance, for the package to reach the ground is
A) 1.0 s
B) 1.5 s
C) 2.0 s
D) 2.5 s
E) 3.0 s

Accepted Answer

The package is dropped with an initial velocity of 16 ft/s downwards from a height of 32 ft.
Using the equation for distance traveled under constant acceleration,
s = ut + (1/2)at^2
where s is the distance traveled, u is the initial velocity, a is the acceleration, and t is the time taken.
Since the package is dropped, the initial velocity u is 0 ft/s.
The acceleration a is equal to the acceleration due to gravity, which is approximately 32 ft/s^2 downwards.
The distance traveled s is equal to the initial height of the package, which is 32 ft downwards.
Substituting in the values, we get:
32 = 0*t + (1/2)*(32)*t^2
Simplifying, we get:
16t^2 = 32
t^2 = 2
t = sqrt(2) seconds
t is approximately 1.4 seconds.
Therefore, the best choice is C) 2.0 s.

Question 17

The graph is a plot of velocity versus time for a moving object during a particular time interval. Which of the following statements is correct?
A) The acceleration of the object is zero.
B) The acceleration of the object is constant.
C) The acceleration of the object is positive and increasing in magnitude.
D) The acceleration of the object is negative and decreasing in magnitude.
E) The acceleration of the object is positive and decreasing in magnitude.

Accepted Answer

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Question 18

Two of the graphs shown are INCORRECT for a particle undergoing one-dimensional motion with constant acceleration. They are
A) 1 and 2
B) 2 and 3
C) 3 and 4
D) 4 and 5
E) 1 and 5

Accepted Answer

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Question 19

A hammer and feather are dropped from the same height above the lunar surface. Which object hits the ground first?
A) the hammer
B) neither because they both float in space
C) the feather
D) both at the same time
E) none of the above

Accepted Answer

On the lunar surface, there is no atmosphere to cause air resistance, so all objects fall at the same rate regardless of their mass. This means a hammer and a feather would hit the ground at the same time.

Question 20

A particle initially at rest undergoes rectilinear (i.e., straight line) motion with an acceleration that is constant in magnitude and direction. The velocity of the particle
A) is constant in magnitude and direction.
B) is constant in direction only.
C) is constant in magnitude only.
D) can change in magnitude and direction.
E) is described by none of these.

Accepted Answer

When a particle undergoes rectilinear motion with constant acceleration, its velocity changes in magnitude but not in direction. The direction of the velocity remains constant along the straight line of motion.

Quiz 2: Motion in One Dimension

The velocity of a particle is given by v(t) = 3t. The average velocity for the particle between t = 2 and 4 s is

The relationship between the velocity of a body moving along the x axis and time is given by v = 2t3 + 2t, where the units are SI units. The total distance the body travels between the times t = 2 s and t = 4 s is

A baseball is thrown vertically up to a height of 30 m on Earth. If the same ball is thrown up on the moon with the same initial speed how much further will it travel up? (Assume gmoon = gearth/6)

An object is thrown upward with a velocity of 32 ft/s from a stationary balloon which is 48 ft above the ground. If air resistance is ignored, the total time until the object impacts the ground is

A ball is thrown upward from an 80-ft tower with an initial vertical speed of 40 ft/s. If air resistance is ignored, the ball's speed when it reaches the ground will be

The change in displacement for a given time interval can be interpreted as

A sandbag is released from a rising air balloon and hits the ground 7 seconds later. From what height was the sandbag dropped from if at the moment of release the balloon was traveling upward at 3 m/s.

Two baseballs are thrown vertically up from the ground at the same speed, one on Earth, and one on Mars. The baseball on Earth reaches a maximum height of 25 m. Which ball hits the ground first and by what time difference? (gMars = 0.38 gEarth)

The graph shows the instantaneous velocity of a car during 15 s of its motion. The distance traveled by this car during this 15-s interval is

The relationship between the velocity of a body moving along the x axis and time is given by v = 3t2 - 2t, where the units are SI units. The total distance the body travels between the times t = 2 s and t = 4 s is

Which of the curves best describes a body moving with a constant non-zero acceleration?

Only one of the following statements is correct. The correct statement is:

The change in velocity for a given time interval can be interpreted as

The graph shows the instantaneous velocity of a car during 15 s of its motion. The distance traveled by this car during this 15-s interval is

A ball has been thrown vertically upward. The graph shows the ball's position as a function of time. Which one of the following statements best describes the motion of the ball?

A balloon is ascending at a rate of 16 ft/s at a height of 32 ft above the ground when a package is dropped. The time taken, in the absence of air resistance, for the package to reach the ground is

The graph is a plot of velocity versus time for a moving object during a particular time interval. Which of the following statements is correct?

Two of the graphs shown are INCORRECT for a particle undergoing one-dimensional motion with constant acceleration. They are

A hammer and feather are dropped from the same height above the lunar surface. Which object hits the ground first?

A particle initially at rest undergoes rectilinear (i.e., straight line) motion with an acceleration that is constant in magnitude and direction. The velocity of the particle

The relationship between the velocity of a body moving along the x axis and time is given by v = 2t³ + 2t, where the units are SI units. The total distance the body travels between the times t = 2 s and t = 4 s is

A baseball is thrown vertically up to a height of 30 m on Earth. If the same ball is thrown up on the moon with the same initial speed how much further will it travel up? (Assume g_moon = g_earth/6)

Two baseballs are thrown vertically up from the ground at the same speed, one on Earth, and one on Mars. The baseball on Earth reaches a maximum height of 25 m. Which ball hits the ground first and by what time difference? (g_Mars = 0.38 g_Earth)

The relationship between the velocity of a body moving along the x axis and time is given by v = 3t² - 2t, where the units are SI units. The total distance the body travels between the times t = 2 s and t = 4 s is