47. Car A moves at a constant speed of $8\,\dfrac{\text{m}}{\text{s}}$ along a straight path, and car B moves behind it at a constant speed of $20\,\dfrac{\text{m}}{\text{s}}$ in the same direction. When the distance between them decreases to 46 meters, car A begins to decelerate with a constant acceleration of $2\,\dfrac{\text{m}}{\text{s}^2}$ and simultaneously car B also decelerates with a constant acceleration of $4\,\dfrac{\text{m}}{\text{s}^2}$. What is the speed of car B at the moment it reaches car A, in meters per second?
(1) $2$ (2) $8$ (3) $4$ (4) $6$

50. The equation of motion of an object with mass $500\,\text{g}$ moving along the $x$-axis is, in SI units, $\vec{F} = (6 - 3t)\hat{i}$. The net average force exerted on this object during the time interval $t_1 = 1\,\text{s}$ to $t_2 = 3\,\text{s}$, in newtons, is:
(1) $3\hat{i}$ (2) $-3\hat{i}$ (3) $6\hat{i}$ (4) $-6\hat{i}$
\hrule
Space for calculations
%% Page 5

44-- The position--time graph of a particle moving along the $x$-axis with constant acceleration is shown below. What is the magnitude of the average speed of the particle in the first 7 seconds?
\begin{minipage}{0.45\textwidth} [Figure: position-time graph with x(m) axis showing value 24 at top, curve passing through origin, reaching a minimum near t=2, then rising to 24 at t=7; t(s) axis shows values 2 and 7] \end{minipage} \begin{minipage}{0.45\textwidth} (1) $\dfrac{25}{8}$
(2) $\dfrac{25}{7}$
(3) $\dfrac{23}{8}$
(4) $\dfrac{23}{7}$ \end{minipage}

46. The figure below shows the acceleration–time graph of a moving object that at moment $t = 0\,\text{s}$ has velocity $\vec{V} = +\!\left(8\,\dfrac{\text{m}}{\text{s}}\right)\hat{i}$ and has been moving. What is the average velocity of the object in these 8 seconds (in meters per second)?
[Figure: acceleration-time graph with $a\,(\frac{\text{m}}{\text{s}^2})$ on vertical axis and $t\,(\text{s})$ on horizontal axis. The graph shows $a = +2$ from $t=0$ to $t=3$, then $a = -6$ from $t=3$ to $t=8$.]

• [(1)] $12$
• [(2)] $15$
• [(3)] $\dfrac{43}{4}$
• [(4)] $\dfrac{53}{6}$

47. An object at moment $t = 0\,\text{s}$ starts moving from rest with constant acceleration. The displacement of this object in the $n$-th second is how many times the displacement in the second $n$ equals the displacement in the second two?
$$\frac{5}{3} \quad (1) \qquad \frac{9}{4} \quad (2) \qquad \frac{3}{2} \quad (3) \qquad 2n \quad (4)$$

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 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) $\sqrt{2s(f + r)}$

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 of mass 0.3 kg is subjected to a force $F = -kx$ with $k = 15 \mathrm{~N}/\mathrm{m}$. What will be its initial acceleration if it is released from a point 20 cm away from the origin?
(1) $3 \mathrm{~m}/\mathrm{s}^2$
(2) $15 \mathrm{~m}/\mathrm{s}^2$
(3) $5 \mathrm{~m}/\mathrm{s}^2$
(4) $10 \mathrm{~m}/\mathrm{s}^2$

Consider a car moving on a straight road with a speed of $100 \mathrm{~m}/\mathrm{s}$. The distance at which car can be stopped is $[\mu_\mathrm{k} = 0.5]$
(1) 800 m
(2) 1000 m
(3) 100 m
(4) 400 m

A bullet fired into a fixed target loses half of its velocity after penetrating 3 cm. How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion?
(1) 3.0 cm
(2) 2.0 cm
(3) 1.5 cm
(4) 1.0 cm

A body is at rest at $x = 0$. At $t = 0$, it starts moving in the positive $x$-direction with a constant acceleration. At the same instant another body passes through $x = 0$ moving in the positive $x$ direction with a constant speed. The position of the first body is given by $\mathrm { x } _ { 1 } ( \mathrm { t } )$ after time ' t ' and that of the second body by $x _ { 2 } ( t )$ after the same time interval. Which of the following graphs correctly describes $\left( x _ { 1 } - x _ { 2 } \right)$ as a function of time ' $t$ '?
(1), (2), (3), (4) [see graphs in original]

Two fixed frictionless inclined plane making an angle $30 ^ { \circ }$ and $60 ^ { \circ }$ with the vertical are shown in the figure. Two block $A$ and $B$ are placed on the two planes. What is the relative vertical acceleration of $A$ with respect to $B$?
(1) $4.9 \mathrm {~ms} ^ { - 2 }$ in horizontal direction
(2) $9.8 \mathrm {~ms} ^ { - 2 }$ in vertical direction
(3) zero
(4) $4.9 \mathrm {~ms} ^ { - 2 }$ in vertical direction

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) $\sqrt { \left( \frac { u ^ { 2 } + v ^ { 2 } } { 2 } \right) }$

The distance travelled by a body moving along a line in time $t$ is proportional to $t^{3}$. The acceleration-time $(a, t)$ graph for the motion of the body will be
(1) [graph 1] (2) [graph 2] (3) [graph 3] (4) [graph 4]

From the top of a 64 metres high tower, a stone is thrown upwards vertically with the velocity of $48 \mathrm {~m} / \mathrm { s }$. The greatest height (in metres) attained by the stone, assuming the value of the gravitational acceleration $g = 32 \mathrm {~m} / \mathrm { s } ^ { 2 }$, is:
(1) 112
(2) 88
(3) 128
(4) 100

A car is standing 200 m behind a bus, which is also at rest. The two start moving at the same instant but with different forward accelerations. The bus has acceleration $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ and the car has acceleration $4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The car will catch up with the bus after time :
(1) $\sqrt { 120 } \mathrm {~s}$
(2) 15 s
(3) $\sqrt { 110 } \mathrm {~s}$
(4) $10 \sqrt { 2 } \mathrm {~s}$

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 bullet of mass 20 g has an initial speed of $1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, just before it starts penetrating a mud wall of thickness 20 cm . If the wall offers a mean resistance of $2.5 \times 10 ^ { - 2 } \mathrm {~N}$, the speed of the bullet after emerging from the other side of the wall is close to:
(1) $0.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $0.1 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A particle starts from the origin at time $t = 0$ and moves along the positive $x$-axis. The graph of velocity with respect to time is shown in figure. What is the position of the particle at time $t = 5s$?
(1) $10 m$
(2) $9 m$
(3) $6 m$
(4) $3 m$