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.

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.

QUESTION 143
A car travels at a constant speed of 90 km/h. The time, in hours, for this car to travel 270 km is
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

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 $85 \mathrm{~s}$.

If a body looses half of its velocity on penetrating 3 cm in a wooden block, then how much will it penetrate more before coming to rest?
(1) 1 cm
(2) 2 cm
(3) 3 cm
(4) 4 cm

Speeds of two identical cars are $u$ and $4u$ at the specific instant. The ratio of the respective distances in which the two cars are stopped from that instant is
(1) $1 : 1$
(2) $1 : 4$
(3) $1 : 8$
(4) $1 : 16$

A car, moving with a speed of $50 \mathrm{~km} / \mathrm{hr}$, can be stopped by brakes after at least 6 m. If the same car is moving at a speed of $100 \mathrm{~km} / \mathrm{hr}$, the minimum stopping distance is
(1) 12 m
(2) 18 m
(3) 24 m
(4) 6 m

A body travels a distance $s$ in $t$ seconds. It starts from rest and ends at rest. In the first part of the journey, it moves with constant acceleration $f$ and in the second part with constant retardation $r$. The value of $t$ is given by
(1) $\sqrt{2s\left(\frac{1}{f} + \frac{1}{r}\right)}$
(2) $2s\left(\frac{1}{f} + \frac{1}{r}\right)$
(3) $\frac{2s}{\frac{1}{f} + \frac{1}{r}}$
(4) $\sqrt{2s(f + r)}$

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $\overrightarrow{\mathrm{u}}$ and the other from rest with uniform acceleration $\overrightarrow{\mathrm{f}}$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle w.r.t. the first is least after a time.
(1) $\frac{u\cos\alpha}{f}$
(2) $\frac{u\sin\alpha}{f}$
(3) $\frac{f\cos\alpha}{u}$
(4) $u\sin\alpha$

A ball is released from the top of a tower of height $h$ metres. It takes $T$ seconds to reach the ground. What is the position of the ball in $\mathrm { T } / 3$ seconds?
(1) $\mathrm { h } / 9$ metres from the ground
(2) $7 \mathrm {~h} / 9$ metres from the ground
(3) $8 \mathrm {~h} / 9$ metres from the ground
(4) $17 \mathrm {~h} / 18$ metres from the ground.

An automobile travelling with speed of $60 \mathrm {~km} / \mathrm { h }$, can brake to stop within a distance of 20 cm . If the car is going twice as fast, i.e $120 \mathrm {~km} / \mathrm { h }$, the stopping distance will be
(1) 20 m
(2) 40 m
(3) 60 m
(4) 80 m

A car starting from rest accelerates at the rate $f$ through a distance $S$, then continues at constant speed for time $t$ and then decelerates at the rate $f/2$ to come to rest. If the total distance traversed is 15 S, then
(1) $S = ft$
(2) $\mathrm{S} = 1/6\, \mathrm{ft}^2$
(3) $\mathrm{S} = 1/2\, \mathrm{ft}^2$
(4) None of these

A parachutist after bailing out falls 50 m without friction. When parachute opens, it decelerates at $2 \mathrm{~m}/\mathrm{s}^2$. He reaches the ground with a speed of $3 \mathrm{~m}/\mathrm{s}$. At what height, did he bail out?
(1) 91 m
(2) 182 m
(3) 293 m
(4) 111 m

Two points $A$ and $B$ move from rest along a straight line with constant acceleration $f$ and $f'$ respectively. If $A$ takes $m$ sec. more than $B$ and describes '$n$' units more than $B$ in acquiring the same speed then
(1) $\left(f - f'\right)m^2 = ff'n$
(2) $\left(f + f'\right)m^2 = ff'n$
(3) $\frac{1}{2}\left(f + f'\right)m = ff'n^2$
(4) $\left(f' - f\right)n = \frac{1}{2}ff'm^2$

A particle has an initial velocity $3 \hat { i } + 4 \hat { j }$ and an acceleration of $0.4 \hat { i } + 0.3 \hat { j }$. Its speed after 10 s is
(1) 10 units
(2) $7 \sqrt { 2 }$ units
(3) 7 units
(4) 8.5 units

A goods train accelerating uniformly on a straight railway track, approaches an electric pole standing on the side of track. Its engine passes the pole with velocity $u$ and the guard's room passes with velocity $v$. The middle wagon of the train passes the pole with a velocity.
(1) $\frac { u + v } { 2 }$
(2) $\frac { 1 } { 2 } \sqrt { u ^ { 2 } + v ^ { 2 } }$
(3) $\sqrt { u v }$
(4) $\sqrt { \left( \frac { u ^ { 2 } + v ^ { 2 } } { 2 } \right) }$

A person climbs up a stalled escalator in 60 s. If standing on the same but escalator running with constant velocity he takes 40 s. How much time is taken by the person to walk up the moving escalator?
(1) 37 s
(2) 27 s
(3) 24 s
(4) 45 s

A particle moves from the point $( 2.0 \hat { i } + 4.0 \hat { j } ) \mathrm { m }$, at $\mathrm { t } = 0$, with an initial velocity $( 5.0 \hat { i } + 4.0 \hat { j } ) \mathrm { ms } ^ { - 1 }$. It is acted upon by a constant force which produces a constant acceleration $( 4.0 \hat { i } + 4.0 \hat { j } ) \mathrm { ms } ^ { - 2 }$. What is the distance of the particle from the origin at time 2 s ?
(1) 15 m
(2) $20 \sqrt { 2 } \mathrm {~m}$
(3) 5 m
(4) $10 \sqrt { 2 } \mathrm {~m}$

A passenger train of length $60 m$ travels at a speed of $80 \mathrm {~km} / \mathrm { hr }$. Another freight train of length $120 m$ travels at a speed of $30 \mathrm {~km} / \mathrm { hr }$. The ratio of times taken by the passenger train to completely cross the freight train when: (i) they are moving in the same direction, and (ii) in the opposite directions is:
(1) $\frac { 5 } { 2 }$
(2) $\frac { 3 } { 2 }$
(3) $\frac { 11 } { 5 }$
(4) $\frac { 25 } { 11 }$

A ball is dropped from the top of a 100 m high tower on a planet. In the last $\frac{1}{2}\,\mathrm{s}$ before hitting the ground, it covers a distance of 19 m. Acceleration due to gravity (in $\mathrm{m\,s^{-2}}$) near the surface on that planet is $\underline{\hspace{1cm}}$

A car accelerates from rest at a constant rate $\alpha$ for some time after which it decelerates at a constant rate $\beta$ to come to rest. If the total time elapsed is t seconds, the total distance travelled is:
(1) $\frac { 4 \alpha \beta } { ( \alpha + \beta ) } \mathrm { t } ^ { 2 }$
(2) $\frac { 2 \alpha \beta } { ( \alpha + \beta ) } t ^ { 2 }$
(3) $\frac { \alpha \beta } { 2 ( \alpha + \beta ) } \mathrm { t } ^ { 2 }$
(4) $\frac { \alpha \beta } { 4 ( \alpha + \beta ) } \mathrm { t } ^ { 2 }$

A ball is thrown up with a certain velocity so that it reaches a height $h$. Find the ratio of the two different times of the ball reaching $\frac { h } { 3 }$ in both the directions.
(1) $\frac { \sqrt { 2 } - 1 } { \sqrt { 2 } + 1 }$
(2) $\frac { 1 } { 3 }$
(3) $\frac { \sqrt { 3 } - \sqrt { 2 } } { \sqrt { 3 } + \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 }$

A scooter accelerates from rest for time $t _ { 1 }$ at constant rate $a _ { 1 }$ and then retards at constant rate $a _ { 2 }$ for time $t _ { 2 }$ and comes to rest. The correct value of $\frac { t _ { 1 } } { t _ { 2 } }$ will be :
(1) $\frac { a _ { 2 } } { a _ { 1 } }$
(2) $\frac { a _ { 1 } } { a _ { 2 } }$
(3) $\frac { a _ { 1 } + a _ { 2 } } { a _ { 1 } }$
(4) $\frac { a _ { 1 } + a _ { 2 } } { a _ { 2 } }$

A small toy starts moving from the position of rest under a constant acceleration. If it travels a distance of 10 m in $t$ s, the distance travelled by the toy in the next $t$ s will be:
(1) 10 m
(2) 20 m
(3) 30 m
(4) 40 m

A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height $h$. Find the ratio of the times in which it is at height $\frac { h } { 3 }$ while going up and coming down respectively.
(1) $\frac { \sqrt { 2 } - 1 } { \sqrt { 2 } + 1 }$
(2) $\frac { \sqrt { 3 } - \sqrt { 2 } } { \sqrt { 3 } + \sqrt { 2 } }$
(3) $\frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 }$
(4) $\frac { 1 } { 3 }$