12. A particle travels in a straight line with a constant acceleration of 3 meters per second per second. If the velocity of the particle is 10 meters per second at time 2 seconds, how far does the particle travel during the time interval when its velocity increases from 4 meters per second to 10 meters per second?
(A) 20 m
(B) 14 m
(C) 7 m
(D) 6 m
(E) 3 m

A race car is traveling on a straight track at a velocity of 80 meters per second when the brakes are applied at time $t = 0$ seconds. From time $t = 0$ to the moment the race car stops, the acceleration of the race car is given by $a ( t ) = - 6 t ^ { 2 } - t$ meters per second per second. During this time period, how far does the race car travel?
(A) 188.229 m
(B) 198.766 m
(C) 260.042 m
(D) 267.089 m

A particle moves along the $X$-axis so that at time $t$ its position is given by $x ( t ) = t ^ { 3 } - 6 t ^ { 2 } + 9 t + 11$. (a) What is the velocity of the particle at $t = 0$ ? (b) During what time intervals is the particle moving to the left? (c) What is the total distance traveled by the particle from $t = 0$ to $t = 2$ ?

A car is traveling on a straight road with velocity $55\,\mathrm{ft/sec}$ at time $t = 0$. For $0 \leq t \leq 18$ seconds, the car's acceleration $a(t)$, in $\mathrm{ft/sec}^{2}$, is the piecewise linear function defined by the graph above.
(a) Is the velocity of the car increasing at $t = 2$ seconds? Why or why not?
(b) At what time in the interval $0 \leq t \leq 18$, other than $t = 0$, is the velocity of the car $55\,\mathrm{ft/sec}$? Why?
(c) On the time interval $0 \leq t \leq 18$, what is the car's absolute maximum velocity, in $\mathrm{ft/sec}$, and at what time does it occur? Justify your answer.
(d) At what times in the interval $0 \leq t \leq 18$, if any, is the car's velocity equal to zero? Justify your answer.

An object moves along the $x$-axis with initial position $x ( 0 ) = 2$. The velocity of the object at time $t \geq 0$ is given by $v ( t ) = \sin \left( \frac { \pi } { 3 } t \right)$.
(a) What is the acceleration of the object at time $t = 4$?
(b) Consider the following two statements. Statement I: For $3 < t < 4.5$, the velocity of the object is decreasing. Statement II: For $3 < t < 4.5$, the speed of the object is increasing. Are either or both of these statements correct? For each statement provide a reason why it is correct or not correct.
(c) What is the total distance traveled by the object over the time interval $0 \leq t \leq 4$?
(d) What is the position of the object at time $t = 4$?

A car is traveling on a straight road. For $0 \leq t \leq 24$ seconds, the car's velocity $v ( t )$, in meters per second, is modeled by the piecewise-linear function defined by the graph above.
(a) Find $\int _ { 0 } ^ { 24 } v ( t ) \, dt$. Using correct units, explain the meaning of $\int _ { 0 } ^ { 24 } v ( t ) \, dt$.
(b) For each of $v ^ { \prime } ( 4 )$ and $v ^ { \prime } ( 20 )$, find the value or explain why it does not exist. Indicate units of measure.
(c) Let $a ( t )$ be the car's acceleration at time $t$, in meters per second per second. For $0 < t < 24$, write a piecewise-defined function for $a ( t )$.
(d) Find the average rate of change of $v$ over the interval $8 \leq t \leq 20$. Does the Mean Value Theorem guarantee a value of $c$, for $8 < c < 20$, such that $v ^ { \prime } ( c )$ is equal to this average rate of change? Why or why not?

Rocket $A$ has positive velocity $v(t)$ after being launched upward from an initial height of 0 feet at time $t = 0$ seconds. The velocity of the rocket is recorded for selected values of $t$ over the interval $0 \leq t \leq 80$ seconds, as shown in the table below.

\begin{tabular}{ c } $t$

(seconds)

& 0 & 10 & 20 & 30 & 40 & 50 & 60 & 70 & 80 \hline

$v(t)$

(feet per second)

& 5 & 14 & 22 & 29 & 35 & 40 & 44 & 47 & 49 \hline \end{tabular}
(a) Find the average acceleration of rocket $A$ over the time interval $0 \leq t \leq 80$ seconds. Indicate units of measure.
(b) Using correct units, explain the meaning of $\int_{10}^{70} v(t)\, dt$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int_{10}^{70} v(t)\, dt$.
(c) Rocket $B$ is launched upward with an acceleration of $a(t) = \frac{3}{\sqrt{t+1}}$ feet per second per second. At time $t = 0$ seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time $t = 80$ seconds? Explain your answer.

A particle moves along the $x$-axis so that its velocity at time $t$, for $0 \leq t \leq 6$, is given by a differentiable function $v$ whose graph is shown above. The velocity is 0 at $t = 0 , t = 3$, and $t = 5$, and the graph has horizontal tangents at $t = 1$ and $t = 4$. The areas of the regions bounded by the $t$-axis and the graph of $v$ on the intervals $[ 0,3 ] , [ 3,5 ]$, and $[ 5,6 ]$ are 8, 3, and 2, respectively. At time $t = 0$, the particle is at $x = - 2$.
(a) For $0 \leq t \leq 6$, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer.
(b) For how many values of $t$, where $0 \leq t \leq 6$, is the particle at $x = - 8$ ? Explain your reasoning.
(c) On the interval $2 < t < 3$, is the speed of the particle increasing or decreasing? Give a reason for your answer.
(d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer.

A car is traveling on a straight road. For $0 \leq t \leq 24$ seconds, the car's velocity $v ( t )$, in meters per second, is modeled by the piecewise-linear function defined by the graph above.
(a) Find $\int _ { 0 } ^ { 24 } v ( t ) d t$. Using correct units, explain the meaning of $\int _ { 0 } ^ { 24 } v ( t ) d t$.
(b) For each of $v ^ { \prime } ( 4 )$ and $v ^ { \prime } ( 20 )$, find the value or explain why it does not exist. Indicate units of measure.
(c) Let $a ( t )$ be the car's acceleration at time $t$, in meters per second per second. For $0 < t < 24$, write a piecewise-defined function for $a ( t )$.
(d) Find the average rate of change of $v$ over the interval $8 \leq t \leq 20$. Does the Mean Value Theorem guarantee a value of $c$, for $8 < c < 20$, such that $v ^ { \prime } ( c )$ is equal to this average rate of change? Why or why not?

A particle moves along a straight line. For $0 \leq t \leq 5$, the velocity of the particle is given by $v ( t ) = - 2 + \left( t ^ { 2 } + 3 t \right) ^ { 6 / 5 } - t ^ { 3 }$, and the position of the particle is given by $s ( t )$. It is known that $s ( 0 ) = 10$.
(a) Find all values of $t$ in the interval $2 \leq t \leq 4$ for which the speed of the particle is 2.
(b) Write an expression involving an integral that gives the position $s ( t )$. Use this expression to find the position of the particle at time $t = 5$.
(c) Find all times $t$ in the interval $0 \leq t \leq 5$ at which the particle changes direction. Justify your answer.
(d) Is the speed of the particle increasing or decreasing at time $t = 4$? Give a reason for your answer.

Particle $P$ moves along the $x$-axis such that, for time $t > 0$, its position is given by $x_P(t) = 6 - 4e^{-t}$. Particle $Q$ moves along the $y$-axis such that, for time $t > 0$, its velocity is given by $v_Q(t) = \dfrac{1}{t^2}$. At time $t = 1$, the position of particle $Q$ is $y_Q(1) = 2$.
(a) Find $v_P(t)$, the velocity of particle $P$ at time $t$.
(b) Find $a_Q(t)$, the acceleration of particle $Q$ at time $t$. Find all times $t$, for $t > 0$, when the speed of particle $Q$ is decreasing. Justify your answer.
(c) Find $y_Q(t)$, the position of particle $Q$ at time $t$.
(d) As $t \to \infty$, which particle will eventually be farther from the origin? Give a reason for your answer.

A car is traveling on a straight road with velocity $55\,\mathrm{ft/sec}$ at time $t = 0$. For $0 \leq t \leq 18$ seconds, the car's acceleration $a(t)$, in $\mathrm{ft/sec}^2$, is the piecewise linear function defined by the graph above.
(a) Is the velocity of the car increasing at $t = 2$ seconds? Why or why not?
(b) At what time in the interval $0 \leq t \leq 18$, other than $t = 0$, is the velocity of the car $55\,\mathrm{ft/sec}$? Why?
(c) On the time interval $0 \leq t \leq 18$, what is the car's absolute maximum velocity, in $\mathrm{ft/sec}$, and at what time does it occur? Justify your answer.
(d) At what times in the interval $0 \leq t \leq 18$, if any, is the car's velocity equal to zero? Justify your answer.

Rocket $A$ has positive velocity $v(t)$ after being launched upward from an initial height of 0 feet at time $t = 0$ seconds. The velocity of the rocket is recorded for selected values of $t$ over the interval $0 \leq t \leq 80$ seconds, as shown in the table below.

$t$ (seconds)	0	10	20	30	40	50	60	70	80
$v(t)$ (feet per second)	5	14	22	29	35	40	44	47	49

(a) Find the average acceleration of rocket $A$ over the time interval $0 \leq t \leq 80$ seconds. Indicate units of measure.
(b) Using correct units, explain the meaning of $\int_{10}^{70} v(t)\, dt$ in terms of the rocket's flight. Use a midpoint Riemann sum with 3 subintervals of equal length to approximate $\int_{10}^{70} v(t)\, dt$.
(c) Rocket $B$ is launched upward with an acceleration of $a(t) = \frac{3}{\sqrt{t+1}}$ feet per second per second. At time $t = 0$ seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time $t = 80$ seconds? Explain your answer.

Caren rides her bicycle along a straight road from home to school, starting at home at time $t = 0$ minutes and arriving at school at time $t = 12$ minutes. During the time interval $0 \leq t \leq 12$ minutes, her velocity $v(t)$, in miles per minute, is modeled by the piecewise-linear function whose graph is shown above.
(a) Find the acceleration of Caren's bicycle at time $t = 7.5$ minutes. Indicate units of measure.
(b) Using correct units, explain the meaning of $\int_{0}^{12} |v(t)| \, dt$ in terms of Caren's trip. Find the value of $\int_{0}^{12} |v(t)| \, dt$.
(c) Shortly after leaving home, Caren realizes she left her calculus homework at home, and she returns to get it. At what time does she turn around to go back home? Give a reason for your answer.
(d) Larry also rides his bicycle along a straight road from home to school in 12 minutes. His velocity is modeled by the function $w$ given by $w(t) = \frac{\pi}{15} \sin\left(\frac{\pi}{12} t\right)$, where $w(t)$ is in miles per minute for $0 \leq t \leq 12$ minutes. Who lives closer to school: Caren or Larry? Show the work that leads to your answer.

A particle moves along a line so that its acceleration for $t \geq 0$ is given by $a ( t ) = \frac { t + 3 } { \sqrt { t ^ { 3 } + 1 } }$. If the particle's velocity at $t = 0$ is 5, what is the velocity of the particle at $t = 3$ ?
(A) 0.713
(B) 1.134
(C) 6.134
(D) 6.710
(E) 11.710

You have been hired to synchronize the four traffic lights on an avenue, indicated by the letters O, A, B, and C, as shown in the figure.
The traffic lights are separated by a distance of 500 m. According to statistical data from the traffic control company, a vehicle that is initially stopped at traffic light O typically starts with constant acceleration of $1 \mathrm{~m~s}^{-2}$ until reaching a speed of $72 \mathrm{~km~h}^{-1}$, and from then on, proceeds at constant speed. You must adjust traffic lights A, B, and C so that they change to green when the vehicle is 100 m away from crossing them, so that it does not have to reduce speed at any moment.
Considering these conditions, approximately how long after the opening of traffic light O should traffic lights A, B, and C open, respectively?
(A) $20 \mathrm{~s}, 45 \mathrm{~s}$ and $70 \mathrm{~s}$.
(B) $25 \mathrm{~s}, 50 \mathrm{~s}$ and $75 \mathrm{~s}$.
(C) $28 \mathrm{~s}, 42 \mathrm{~s}$ and $53 \mathrm{~s}$.
(D) $30 \mathrm{~s}, 55 \mathrm{~s}$ and $80 \mathrm{~s}$.
(E) $35 \mathrm{~s}, 60 \mathrm{~s}$ and $85 \mathrm{~s}$.

Starting from point A, one travels to point B which is 6 km away, and then returns to point A along the same route. For the first 1 km, one walks at a constant speed, and for the remaining 5 km, one travels at twice the initial walking speed. On the return trip, one travels at a speed 2 km/h faster than the initial walking speed. When the total time for the round trip is at most 2 hours 30 minutes, what is the minimum value of the initial walking speed? (Given that the unit of speed is km/h.) [3 points]
(1) $\frac { 12 } { 5 }$
(2) $\frac { 13 } { 5 }$
(3) $\frac { 14 } { 5 }$
(4) 3
(5) $\frac { 16 } { 5 }$

There is a point P that starts from the origin at time $t = 0$ and moves on a number line. For a real number $k$, the velocity $v ( t )$ of point P at time $t$ ($t \geq 0$) is $$v ( t ) = t ^ { 2 } - k t + 4$$ Which of the following in are correct? [4 points]
ᄀ. If $k = 0$, then the position of point P at time $t = 1$ is $\frac { 13 } { 3 }$. ㄴ. If $k = 3$, then the direction of motion of point P changes once after departure. ㄷ. If $k = 5$, then the distance traveled by point P from time $t = 0$ to $t = 2$ is 3.
(1) ᄀ
(2) ᄀ, ᄂ
(3) ᄀ, ᄃ
(4) ㄴ, ㄱ
(5) ᄀ, ᄂ, ᄃ

157- Train A, with length 200 m, is moving at constant speed $4\,\dfrac{\text{m}}{\text{s}}$. Train B, with length 225 m, has stopped on the adjacent track. At the moment train A completely passes train B, train B starts moving in the same direction as train A with constant acceleration $2\,\dfrac{\text{m}}{\text{s}^2}$ and brings its speed up to $50\,\dfrac{\text{m}}{\text{s}}$ and continues at that speed. How many seconds after the start of motion does train B completely pass train A?
(1) $57.5$ (2) $82.5$ (3) $80$ (4) $105$

179- A stone is thrown vertically upward from height $h$ with initial velocity $V_0$ and hits the ground after $4$ seconds. If in the last second of its motion it travels $\dfrac{h}{2}$, how many meters is $h$? $\left(g = 10\,\dfrac{\text{m}}{\text{s}^2}\right)$

• [(1)] $60$
• [(2)] $90$
• [(3)] $120$
• [(4)] $180$

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156. A particle starts from rest at the origin on the $x$-axis with constant acceleration, and at moment $t = 5\text{s}$ reaches position $x = -122.5\,\text{m}$. How many meters per second does the speed of the particle reach at this moment?
(1) $19.6$ (2) $33.4$ (3) $45.0$ (4) $49.0$

160. A driver with a car of mass 2 tons, moving at speed $36\,\dfrac{\text{km}}{\text{h}}$ on a straight horizontal road, applies the brakes upon seeing a red light. The car stops after traveling $4\,\text{m}$. How many Newtons is the braking friction force applied to the car?
(1) $7500$ (2) $12500$ (3) $15000$ (4) $25000$
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42. A bullet is fired vertically upward from the Earth's surface and reaches a height of 42 meters above the Earth's surface, and its kinetic energy decreases by 30\%. What is the maximum height (in meters) this bullet reaches above the Earth's surface?
(air resistance is negligible and $g = 10\,\dfrac{\mathrm{m}}{\mathrm{s}^2}$)
(1) 96 (2) 120 (3) 140 (4) 149

45. On a straight path, starting from a point, object A moves with constant acceleration $a$ from rest, and at $t = 2\,\mathrm{s}$, object B starts from the same point and moves with constant acceleration $a = 0.5\,\dfrac{\mathrm{m}}{\mathrm{s}^2}$ from rest. If at $t = 6\,\mathrm{s}$ the two objects meet each other, what is their distance at $t = 10\,\mathrm{s}$ in meters?
(1) $4.4$ (2) $8.8$ (3) $13.2$ (4) $24.8$

46. A bullet is released from a point 100 meters above the ground. One second later, another bullet is released from the same point. At the moment the first bullet reaches the ground, what change does the distance between the two bullets undergo? (Assume air resistance is negligible.)
(1) remains constant. (2) increases.
(3) decreases. (4) first decreases and then increases.
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