Chapter 2

An automobile travelling with a speed of \(60 \mathrm{kmh}\) can brake to stop with a distance of \(20 \mathrm{~m}\). If the car is going twice as fast i.e., \(120 \mathrm{kms}^{-1}\), the stopping distance will be (a) \(20 \mathrm{~m}\) (b) \(40 \mathrm{~m}\) (c) \(60 \mathrm{~m}\) (d) \(80 \mathrm{~m}\)

A wheel of radius \(1 \mathrm{~m}\) rolls forward half a revolution on a horizontal ground. The magnitude of the displacement of the point of the wheel initially in contact with the ground is (a) \(2 \pi\) (b) \(\sqrt{2} \pi\) (c) \(\sqrt{\pi^{2}+4}\) (d) \(\pi\)

Two balls of same size but the density of one is greater than that of the other are dropped from the same height, then which ball will reach the earth first (air resistance is negligible)? (a) Heavy ball (b) Light ball (c) Both simultaneously (d) Will depend upon the density of the balls

A point particle starting from rest has a velocity that increases linearly with time such that \(v=p t\), where \(p=4 \mathrm{~ms}^{-2}\). The distance covered in the first 2 s will be (a) \(6 \mathrm{~m}\) (b) \(4 \mathrm{~m}\) (c) \(8 \mathrm{~m}\) (d) \(10 \mathrm{~m}\)

A boat takes two hours to travel \(8 \mathrm{~km}\) and back in still water. If the velocity of water \(4 \mathrm{kmh}^{-1}\), the time taken for going ups tream \(8 \mathrm{~km}\) and coming back is (a) \(2 \underline{\mathrm{h}}\) (b) \(2 \mathrm{~h} 40 \mathrm{~min}\) (c) 1 h \(20 \mathrm{~min}\) (d) cannot be estimated with the information given

A particle moving in a straight line covers half the distance with speed of \(3 \mathrm{~m} / \mathrm{s}\). The other half of the distance is covered in two equal time intervals and with speeds of \(4.5 \mathrm{~m} / \mathrm{s}\) and \(7.5 \mathrm{~m} / \mathrm{s}\) respectively. The average speed of the particle during this motion is (a) \(4.0 \mathrm{~m} / \mathrm{s}\) (b) \(5.0 \mathrm{~m} / \mathrm{s}\) (c) \(5.5 \mathrm{~m} / \mathrm{s}\) (d) \(4.8 \mathrm{~m} / \mathrm{s}\)

In a race for \(100 \mathrm{~m}\) dash, the first and the second runners have a gap of one metre at the mid way stage. Assuming the first runner goes steady, by what percentage should the second runner increases his speed just to win the race. (a) \(2 \%\) (b) \(4 \%\) (c) more than \(4 \%\) (d) less than 496

The displacement of a body along \(x\)-axis depends on time as \(\sqrt{x}=t+1\). Then, the velocity of body (a) increase with time (b) decrease with time (c) independent of time (d) None of these

Two cars \(A\) and \(B\) are travelling in the same direction with velocities \(v_{A}\) and \(v_{B}\left(v_{A}>v_{B}\right) .\) When the car \(A\) is at a distance \(s\) behind car \(B\), the driver of the car \(A\) applies the brakes producing a uniform retardation a, there will be no collision when (a) \(s<\frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\) (b) \(s=\frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\) (c) \(s \geq \frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\) (d) \(s \leq \frac{\left(v_{A}-v_{B}\right)^{2}}{2 a}\)

A car moving along a straight highway with speed of \(126 \mathrm{~km} / \mathrm{h}\) is brought to a stop within a distance of \(200 \mathrm{~m}\). What is the retardation of the car (assumed uniform) and how long does it take for the car to stop? [NCERT] (a \(3.06 \mathrm{~m} / \mathrm{s}^{2}\) and \(11.4 \mathrm{~s}\) (b) \(206 \mathrm{~m} / \mathrm{s}^{2}\) and \(11.4 \mathrm{~s}\) (c) \(3.06 \mathrm{~m} / \mathrm{s}^{2}\) and \(10.4 \mathrm{~s}\) (d) \(3.06 \mathrm{~m} / \mathrm{s}^{2}\) and \(4.1 \mathrm{~s}\)

A bird flies for \(4 \mathrm{~s}\) with a velocity of \(|t-2| \mathrm{ms}^{-1}\) in a straight line, where \(t=\) time in second. It covers a distance of (a) \(8 \mathrm{~m}\) (b) \(6 \mathrm{~m}\) (c) \(4 \mathrm{~m}\) (d) \(2 \mathrm{~m}\)

A body starts from rest and moves with a constant acceleration. The ratio of distance covered in the \(n\)th second to the distance covered in \(n\) second is (a) \(\frac{2}{n}-\frac{1}{n^{2}}\) (b) \(\frac{1}{n^{2}}-\frac{1}{n}\) (c) \(\frac{2}{n^{2}}-\frac{1}{n}\) (d) \(\frac{2}{n}+\frac{1}{n^{2}}\)

A man throws balls with the same speed vertically upwards one after the other at an interval of \(2 \mathrm{~s}\). What should be the speed of the throw so that more than two balls are in the sky at any time? (Given \(g=9.8 \mathrm{~m} / \mathrm{s}^{2}\) ) (a) At least \(0.8 \mathrm{~m} / \mathrm{s}\) (b) Any speed less than \(19.6 \mathrm{~m} / \mathrm{s}\) (c) Only with speed \(19.6 \mathrm{~m} / \mathrm{s}\) (d) More than \(19.6 \mathrm{~m} / \mathrm{s}\)

A particle moving with a uniform acceleration along a straight line covers distance \(a\) and \(b\) in successive intervals of \(p\) and \(q\) second. The acceleration of the particle is (a) \(\frac{p q(p+q)}{2(b p-a q)}\) (b) \(\frac{2(a q-b p)}{p q(p-q)}\) (c) \(\frac{b p-a q}{p q(p-q)}\) (d) \(\frac{2(b p-a q)}{p q(p-q)}\)

A motion boat covers a given distance in \(6 \mathrm{~h}\) moving down stream of a river. It covers the same distance in \(10 \mathrm{~h}\) moving upstream. The time (in hour) it takes to cover the same distance in still water is (a) \(6 \mathrm{~h}\) (b) \(7.5 \mathrm{~h}\) (c) \(10 \mathrm{~h}\) (d) \(15 \mathrm{~h}\)

42\. A particle moves along \(x\)-axis as $$ x=4(t-2)+a(t-2)^{2} $$ Which of the following is true? (a) The initial velocity of particle is 4 (b) The acceleration of particle is \(2 \mathrm{a}\) (c) The particle is at origin at \(t=0\) (d) None of the above

A point initially at rest moves along \(x\)-axis. Its acceleration varies with time as \(a=(6 t+5) \mathrm{m} / \mathrm{s}^{2} .\) If it starts from origin, the distance covered in \(2 \mathrm{~s}\) is (a) \(20 \mathrm{~m}\) (b) \(18 \mathrm{~m}\) (c) \(16 \mathrm{~m}\) (d) \(25 \mathrm{~m}\)

A \(2 \mathrm{~m}\) wide truck is moving with a uniform speed \(v_{0}=8 \mathrm{~ms}^{-1}\) along a straight horizontal road. \(\mathrm{A}\) pedestrian starts to cross the road with a uniform speed \(v\) when the truck is \(4 \mathrm{~m}\) away from him. The minimum value of \(v\), so that he can cross the road safely is \(\begin{array}{ll}\text { (a) } 2.62 \mathrm{~ms}^{-1} & \text { (b) } 4.6 \mathrm{~ms}^{-1}\end{array}\) (c) \(3.57 \mathrm{~ms}^{-1}\) (d) \(1.414 \mathrm{~ms}^{-1}\)

From the top of a tower of height \(50 \mathrm{~m}\), a ball is thrown vertically upwards with a certain velocity. It hits the ground \(10 \mathrm{~s}\) after it is thrown up. How much time does it take to cover a distance \(A B\) where \(A\) and \(B\) are two points \(20 \mathrm{~m}\) and \(40 \mathrm{~m}\) below the edge of the tower? \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(2.0 \mathrm{~s}\) (b) \(1.0 \mathrm{~s}\) (c) \(0.5 \mathrm{~s}\) (d) \(0.4 \mathrm{~s}\)

A bus moves over a straight level road with a constant acceleration \(a .\) A body in the bus drops a ball outside. The acceleration of the ball with respect to the bus and the earth are respectively (a) \(a\) and \(g\) (b) \(a+g\) and \(g-a\) (c) \(\sqrt{a^{2}+g^{2}}\) and \(g\) (d) \(\sqrt{a^{2}+g^{2}}\) and \(a\)

The acceleration of a particle is increasing linearly with time \(t\) as \(b t\). The particle starts from the origin with an initial velocity \(v_{0}\). The distance travelled by the particle in time \(t\), is (a) \(v_{0} t+\frac{1}{3} b t^{2}\) (b) \(v_{0} t+\frac{1}{6} b t^{3}\) (c) \(v_{0} f+\frac{1}{3} b t^{3}\) (d) \(v_{0} t+\frac{1}{2} b t^{2}\)

A particle starts from the origin and moves along the \(X\)-axis such that the velocity at any instant is given by \(4 t^{3}-2 t\), where \(t\) is in second and velocity is in \(\mathrm{ms}^{-1} .\) What is the acceleration of the particle when it is \(2 \mathrm{~m}\) from the origin? (a) \(10 \mathrm{~ms}^{-2}\) (b) \(12 \mathrm{~ms}^{-2}\) (c) \(22 \mathrm{~ms}^{-2}\) (d) \(28 \mathrm{~ms}^{-2}\)

The motion of a body is given by the equation \(\frac{d v(t)}{d t}=6.0-3 v(t)\), where \(v(t)\) is speed in \(\mathrm{ms}^{-1}\) and \(t\) in second. If body was at rest at \(t=0\) (a) the terminal speed is \(2.0 \mathrm{~ms}^{-1}\) (b) the speed varies with the times as \(v(t)=2\left(1-e^{-3 t}\right) \mathrm{ms}^{-1}\) (c) the speed is \(1.0 \mathrm{~ms}^{-1}\) when the acceleration is half the initial value (d) the magnitude of the initial acceleration is \(6.0 \mathrm{~ms}^{-2}\)

The retardation experienced by a moving motor boat, after its engine is cut- off, is given by \(\frac{d v}{d t}=-k v^{3}\), where \(k\) is a constant. If \(v_{0}\) is the magnitude of the velocity at cut-off, the magnitude of the velocity at time \(t\) after the cut-off is (a) \(v_{0}\) (b) \(\frac{v_{0}}{2}\) (c) \(v_{0} e^{-k t}\) (d) \(\frac{v_{0}}{\sqrt{2 v_{0}^{2} k t+1}}\)

An elevator ascends with an upward acceleration of \(2.0 \mathrm{~ms}^{-2}\). At the instant its upward speed is \(2.5 \mathrm{~ms}^{-1}\), loose bolt is dropped from the ceiling of the elevator \(3.0 \mathrm{~m}\) from the floor. If \(g=10 \mathrm{~ms}^{-2}\), then (a) the time of flight of the bolt from the ceiling to floor of the elevator is \(0.11 \mathrm{~s}\) (b) the displacement of the bolt during the free fall relative to the elevator shaft is \(0.75 \mathrm{~m}\) (c) the distance covered by the bolt during the free fall relative to the elevator shaft is \(1.38 \mathrm{~m}\) (d) the distance covered by the bolt during the free fall relative to the elevator shaft is \(2.52 \mathrm{~m}\)

The driver of a car moving with a speed of \(10 \mathrm{~ms}^{-1}\) sees a red light ahead, applies brakes and stops after covering \(10 \mathrm{~m}\) distance. If the same car were moving with a speed of \(20 \mathrm{~ms}^{-1}\), the same driver would have stopped the car after covering \(30 \mathrm{~m}\) distance. Within what distance the car can be stopped if travelling with a velocity of \(15 \mathrm{~ms}^{-1}\) ? Assume the same reaction time and the same deceleration in each case. (a) \(18.75 \mathrm{~m}\) (b) \(20.75 \mathrm{~m}\) (c) \(22.75 \mathrm{~m}\) (d) \(25 \mathrm{~m}\)

A particle of mass \(m\) moves on the \(x\)-axis as follows : it starts from rest at \(t=0\) from the point \(x=0\) and comes to rest at \(t=1\) at the point \(x=1\). No other information is available about its motion at intermediate time \((0

A lift is coming from \(8^{\text {th }}\) floor and is just about to reach \(4^{\text {th }}\) floor. Taking ground floor as origin and positive direction upwards for all quantities, which one of the following is correct? [NCERT Exemplar] (a) \(x<0, v<0, a>0\) (b) \(x>0, v<0, a<0\) (c) \(x>0, v<0, a>0\) (d) \(x>0, v>0, a<0\)

The motion of a body falling from rest in a resisting medium is described by the equation $$ \frac{d v}{d t}=A-B v $$ where \(A\) and \(B\) are constants. Then (a) initial acceleration of the body is \(A\) (b) the velocity at which acceleration becomes zero is \(A / B\) (c) the velocity at any time \(t\) is \(\frac{A}{B}\left(1-e^{B t}\right)\) (d) All of the above are wrong

If the velocity \(v\) of a particle moving along a straight line decreases linearly with its displacement \(s\) from \(20 \mathrm{~ms}^{-1}\) to a value approaching zero at \(s=30 \mathrm{~m}\), then acceleration of the particle at \(s=15 \mathrm{~m}\) is (a) \(\frac{2}{3} \mathrm{~ms}^{-2}\) (b) \(-\frac{2}{3} \mathrm{~ms}^{-2}\) (c) \(\frac{20}{3} \mathrm{~ms}^{-2}\) (d) \(-\frac{20}{3} \mathrm{~ms}^{-2}\)

A spring with one end attached to a mass and the other to a rigid support is stretched and released [NCERT Exemplar] (a) Magnitude of acceleration, when just released is maximum (b) Magnitude of acceleration, when at equilibrium position is maximum (c) Speed is maximum when mass is at equilibrium position (d) Magnitude of displacement is always maximum whenever speed is minimum

The displacement \((x)\) of a particle depends on time \((t)\) as $$ x=\alpha t^{2}-\beta t^{3} $$ (a) The particle will come to rest after time \(2 \alpha / 3 \beta\) (b) The particle will return to its starting point after time \(\alpha / \beta\) (c) No net force will act on the particle at \(t=\alpha / 3 \beta\) (d) The initial velocity of the particle was zero but its initial acceleration was not zero

A police van moving on a highway with a speed of \(30 \mathrm{~km} / \mathrm{h}\) fires a bullet at a thief's car speeding away in the same direction with a speed of \(192 \mathrm{~km} / \mathrm{h}\). If the muzzle speed of the bullet is \(150 \mathrm{~m} / \mathrm{s}\), with what speed does the bullet hit the thief's car? (Note Obtain that speed which is relevant for damaging the thiefs car. [NCERT] (a) \(105 \mathrm{~m} / \mathrm{s}\) (b) \(100 \mathrm{~m} / \mathrm{s}\) (c) \(95 \mathrm{~m} / \mathrm{s}\) (d) \(110 \mathrm{~m} / \mathrm{s}\)

A particle starts from rest and travels a distance \(s\) with uniform acceleration, then it travels a distance \(2 s\) with uniform speed, finally it travels a distance \(3 s\) with uniform retardation and comes to rest. If the complete motion of the particle in a straight line then the ratio of its average velocity to maximum velocity in (a) \(6 / 7\) (b) \(4 / 5\) (c) \(3 / 5\) (d) \(2 / 5\)

A particle is projected vertically upwards in vacuum with a speed \(v\). (a) The time taken to rise to half its maximum height is half the time taken to reach its maximum height (b) The time taken to rise to three-fourth of its maximum height is half the time taken to reach its maximum height (c) When it rises to half its maximum height, its speed becomes \(v / \sqrt{2}\) (d) When it rises to half its maximum height, its speed becomes \(v / 2\)

A particle moving in a straight line with uniform acceleration is observed to be a distance \(a\) from a fixed point initially. It is at distances \(b, c, d\) from the same point after \(n, 2 n, 3 n\) second. The acceleration of the particle is (a) \(\frac{c-2 b+a}{n^{2}}\) (b) \(\frac{c+b+a}{9 n^{2}}\) (c) \(\frac{c+2 b+a}{4 n^{2}}\) (d) \(\frac{c-b+a}{n^{2}}\)

A particle is moving with a uniform acceleration along a straight line \(A B\). Its speed at \(A\) and \(B\) are \(2 \mathrm{~ms}^{-1}\) and \(14 \mathrm{~ms}^{-1}\) respectively. Then (a) its speed at mid-point of \(A B\) is \(10 \mathrm{~ms}^{-1}\) (b) its speed at a point \(P\) such that \(A P: P B=1: 5\) is \(4 \mathrm{~ms}^{-1}\) (c) the time to go from \(A\) to mid-point of \(A B\) is double of that to go from mid-point to \(B\) (d) None of the above

A body is moving along a straight line path with constant velocity. At an instant of time the distance of time the distance travelled by it is \(s\) and its displacement is \(D\), then (a) \(D5\) (c) \(D=s\) (d) \(D \leq s\)

Three particles start from the origin at the same time, one with a velocity \(v_{1}\) along \(x\)-axis, the second along the \(y\)-axis with a velocity \(v_{2}\) and the third along \(x=y\) line. The velocity of the third so that the three may always lie on the same line is (a) \(\frac{v_{1} v_{2}}{v_{1}+v_{2}}\) (b) \(\frac{\sqrt{2} v_{1} v_{2}}{v_{1}+v_{2}}\) (c) \(\frac{\sqrt{3} v_{1} v_{2}}{v_{1}+v_{2}}\) (d) zero

In one dimensional motion, instantaneous speed \(v\) satisfies \(0 \leq v

The engine of a train can impart a maximum acceleration of \(1 \mathrm{~ms}^{-2}\) and the brakes can give a maximum retardation of \(3 \mathrm{~ms}^{-2} .\) The least time during which a train can go from one place to the other place at a distance of \(1.2 \mathrm{~km}\) is nearly (a) \(108 \leq\) (b) \(191 \mathrm{~s}\) (c) \(56.6 \mathrm{~s}\) (d) time is fixed

The acceleration of a particle increasing linearly with time \(t\) is \(b t\). The particle starts from the origin with an initial velocity \(v_{0}\). The distance travelled by the particle in time \(t\) will be (a) \(v_{0} t+\frac{1}{6} b t^{3}\) (b) \(v_{0} t+\frac{1}{6} b t^{2}\) (c) \(v_{0} t+\frac{1}{3} b t^{3}\) (d) \(v_{0} t+\frac{1}{3} b t^{2}\)

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion A body is dropped from a height of \(40.0 \mathrm{~m}\). Afterit falls by half the distance, the acceleration due to gravity ceases to act. The velocity with which it hits the ground is \(20 \mathrm{~ms}^{-1}\) (Take \(g=10 \mathrm{~ms}^{-2}\) ). Reason \(v^{2}=u^{2}+2 a s\)

The displacement-time graphs of two moving particles make angles of \(30^{\circ}\) and \(45^{\circ}\) with the \(x\)-axis. The ratio of the two velocities is (a) \(\sqrt{3}: 1\) (b) \(1: 1\) (c) \(1: 2\) (d) \(1: \sqrt{3}\)

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion The average velocity of the object over an interval of time is either smaller than or equal to the average speed of the object over the same interval. Reason Velocity is a vector quantity and speed is a scalar quantity.

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion The slope of displacement-time graph of a body movng with high velocity is steeper than the slope of displacement-time graph of a body with low velocity. Reason Slope of displacement-time graph = Velocity of the body.

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choice, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given below (a) If both Assertion and Reason are true and the Reason is correct explanation of the Assertion (b) If both Assertion and Reason are true but Reason is not correct explanation of the Assertion (c) If Assertion is true but Reason is false (d) If Assertion is false but the Reason is true Assertion A body having non-zero acceleration can have a constant velocity. Reason Acceleration is the rate of change of velocity.

A ball is dropped from a bridge at a height of \(176.4 \mathrm{~m}\) over a river. After \(2 \mathrm{~s}\), a second ball is thrown straight downwards. What should be the initial velocity of the second ball so that both hit the water simultaneously? (a) \(2.45 \mathrm{~ms}^{-1}\) (b) \(49 \mathrm{~ms}^{-1}\) (c) \(14.5 \mathrm{~ms}^{-1}\) (d) \(24.5 \mathrm{~ms}^{-1}\)

A scooterist sees a bus \(1 \mathrm{~km}\) ahead of him moving with a velocity of \(10 \mathrm{~ms}^{-1}\). With what speed the scooterist should move so as to overtake the bus in \(\begin{array}{ll}100 \mathrm{~s} . ? & \text { [Orissa JEE 2008] }\end{array}\) (a) \(10 \mathrm{~ms}^{-1}\) (b) \(20 \mathrm{~ms}^{-1}\) (c) \(50 \mathrm{~ms}^{-1}\) (d) \(30 \mathrm{~ms}^{-1}\)

A bullet emerges from a barrel of length \(1.2 \mathrm{~m}\) with a speed of \(640 \mathrm{~ms}^{-1}\). Assuming constant acceleration, the approximate time that it spends in the barrel after the gun is fired is [WB JEE 2008] (a) \(4 \mathrm{~ms}\) (b) \(40 \mathrm{~ms}\) (c) \(400 \mathrm{~ms}\) (d) \(1 \mathrm{~s}\)