Chapter 2

A ball \(A\) is thrown up vertically with a speed \(u\) and at the same instant another ball \(B\) is released from a height \(h\). At time \(t\), the speed of \(A\) relative to \(B\) is (a) \(y\) (b) \(2 u\) (c) \(u-g t\) (d) \(\sqrt{\left(u^{2}-g t\right)}\)

A body is fired vertically upwards. At half the maximum height, the velocity of the body is \(10 \mathrm{~ms}^{-1}\). The maximum height raised by the body is \(\begin{array}{ll}\left(g=10 \mathrm{~ms}^{-2}\right) . & \text { [Orissa JEE 2008] }\end{array}\) (a) zero (b) \(10 \mathrm{~m}\) (c) \(15 \mathrm{~m}\) (d) \(20 \mathrm{~m}\)

A body falls freely from rest. It covers as much distance in the last second of its motion as covered in the first three seconds. The body has fallen for a time of (a) \(3 \mathrm{~s}\) (b) \(5 \mathrm{~s}\) (c) \(7 \underline{5}\) (d) \(9 \mathrm{~s}\)

The velocity of a particle is \(v=v_{0}+g t+f t^{2} .\) If its position is \(x=0\) at \(t=0\), then its displacement after unit time \((t=1)\) is (a) \(v_{0}-g / 2+f\) (b) \(v_{0}+g / 2+3 f\) (c) \(v_{0}+g / 2+f / 3\) (d) \(v_{0}+g+f\)

Rain is falling vertically with a speed of \(30 \mathrm{~m} / \mathrm{s} . \mathrm{A}\) woman rides a bicycle with a speed of \(10 \mathrm{~m} / \mathrm{s}\) in the north to south direction. What is the direction in which she should hold her umbrella? [NCERT] (a) \(18^{\circ}\) with vertical (b) \(18^{\circ}\) with horizontal (c) \(28^{\circ}\) with vertical (d) \(28^{\circ}\) with horizontal

A proton in a cyclotron changes its velocity from \(30 \mathrm{kms}^{-1}\) due north to \(40 \mathrm{kms}^{-1}\) due east in \(20 \mathrm{~s}\). What is the magnitude of average acceleration during this time? (a) \(2.5 \mathrm{kms}^{-2}\) (b) \(12.5 \mathrm{kms}^{-2}\) (c) \(22.5 \mathrm{kms}^{-2}\) (d) \(32.5 \mathrm{kms}^{-2}\)

A ball \(P\) is dropped vertically and another ball \(Q\) is thrown horizontally with the same velocities from the same height and at the same time. If air resistance is neglected, then (a) ball \(P\) reaches the ground first (b) ball \(Q\) reaches the ground first (c) both reach the ground at the same time (d) the respective masses of the two balls will decide the time

A body of mass \(m\) is accelerated uniformly from rest to a speed \(v\) in a time \(T\). The instantaneous power delivered to the body as a function of time is given by [AIEEE 2008] (a) \(\frac{1}{2} \frac{m v^{2}}{T^{2}} t^{2}\) (b) \(\frac{1}{2} \frac{m v^{2}}{T^{2}} t\) (c) \(\frac{m v^{2}}{\tau^{2}} t^{2}\) (d) \(\frac{m v^{2}}{T^{2}} t\)

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 body moves with initial velocity \(10 \mathrm{~ms}^{-1}\). If it covers a distance of \(20 \mathrm{~m}\) in \(2 \mathrm{~s}\) then acceleration of the body is [Orissa JEE 2011] (a) zero (b) \(10 \mathrm{~ms}^{-2}\) (c) \(5 \mathrm{~ms}^{-2}\) (d) \(2 \mathrm{~ms}^{-2}\)

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement \(x\) is proportional to [UP SEE 2007] (a) \(x^{2}\) (b) \(\mathrm{e}^{x}\) (c) \(x\) (d) \(\log e^{x}\)

A ball which is at rest, is dropped from a height \(h\) metre. As it bounces off the floor its speed is \(80 \%\) of what it was just before touching the ground? The ball \(\begin{array}{ll}\text { will then rise to nearly a height } & \text { [BVP Engg. 2007] }\end{array}\) (a) \(0.94 \mathrm{~h}\) (b) \(0.80 h\) (c) \(0.75 h\) (d) \(0.64 h\)

A jet airplane travelling at a speed of \(500 \mathrm{~km} / \mathrm{h}\) ejects its products of combustion at the speed of \(1500 \mathrm{~km} / \mathrm{h}\) relative to the jet plane. What is the speed of the latter with respect to an observer on the ground? [NCERT] (a) \(-1000 \mathrm{~km} / \mathrm{h}\) (b) \(1000 \mathrm{~km} / \mathrm{h}\) (c) \(100 \mathrm{~km} / \mathrm{h}\) (d) \(-100 \mathrm{~km} / \mathrm{h}\)

A particle has an initial velocity of \(3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}\) and acceleration of \(0.4 \hat{\mathbf{i}}+0.3 \hat{\mathbf{j}}\). Its speed after \(10 \mathrm{~s}\) is (a) 10 units (b) \(7 \sqrt{2}\) units (c) 7 units (d) \(8.5\) units

A body is thrown vertically up with a velocity \(u\). It passes three points \(A, B\) and \(C\) in its upward journey with velocities \(\frac{u}{2}, \frac{u}{3}\) and \(\frac{u}{4}\) repectively. The ratio of the separations between points \(A\) and \(B\) and between \(B\) and \(C\), i.e.,\(\frac{A B}{B C}\) is \(\begin{array}{llll}\text { (a) } 1 & \text { (b) } 2 & \text { (c) } \frac{10}{7} & \text { (d) } \frac{20}{7}\end{array}\)

A particle is moving with velocity \(\mathbf{v}=k(4 \hat{\mathbf{i}}+x \hat{\mathbf{j}})\) where \(k\) is a constant. The general equation for its path is (a) \(y=x^{2}+\) constant (b) \(y^{2}=x+\) constant (c) \(x y=\) constant (d) \(y^{2}=x^{2}+\) constant

A boy released a ball from the top of a building. It will clear a window \(2 \mathrm{~m}\) high at a distance \(10 \mathrm{~m}\) below the top in nearly (a) \(1 \mathrm{~s}\) (b) \(1.3 \mathrm{~s}\) (c) \(0.6 \mathrm{~s}\) (d) \(0.13 \mathrm{~s}\)

A particle located at \(x=0\) at time \(t=0\), starts moving along the positive \(x\)-direction with a velocity \(v\) that varies as \(v=a \sqrt{x}\). The displacement of the particle varies with time as (a) \(t^{3}\) (b) \(t^{2}\) (c) \(t\) (d) \(t^{1 / 2}\)

A stone is allowed to fall from the top of a tower \(100 \mathrm{~m}\) high and at the same time another stone is projected vertically upwards from the ground with a velocity of \(254 \mathrm{~ms}^{-1}\). The two stones will meet after \(\begin{array}{llll}\text { (a) } 4 \mathrm{~s} & \text { (b) } 0.4 \mathrm{~s} & \text { (c) } 0.04 \mathrm{~s} & \text { (d) } 40 \mathrm{~s}\end{array}\)

An object, moving with a speed of \(6.25 \mathrm{~m} / \mathrm{s}\), is declerated at a rate given by \(\frac{d v}{d t}=-2.5 \sqrt{v}\), where \(v\) is the instantaneous speed. The time taken by the object, to come to rest would be (a) \(2 \mathrm{~s}\) (b) \(4 \mathrm{~s}\) (c) \(8 \mathrm{~s}\) (d) \(1 \mathrm{~s}\)

From a balloon rising vertically upwards at \(5 \mathrm{~m} / \mathrm{s}\) a stone is thrown up at \(10 \mathrm{~m} / \mathrm{s}\) relative to the balloon. Its velocity with respect to ground after \(2 \mathrm{~s}\) is (assume \(\left.g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (a) \(\underline{0}\) (b) \(20 \mathrm{~m} / \mathrm{s}\) (c) \(10 \mathrm{~m} / \mathrm{s}\) (d) \(5 \mathrm{~m} / \mathrm{s}\)

A train accelerated uniformly from rest attains a maximum speed of \(40 \mathrm{~ms}^{-1}\) in \(20 \mathrm{~s}\). It travels at this speed for \(20 \mathrm{~s}\) and is brought to rest with uniform retardation in \(40 \mathrm{~s}\). The average velocity during this period is (a) \((80 / 3) \mathrm{ms}^{-1}\) (b) \(30 \mathrm{~ms}^{-1}\) (c) \(25 \mathrm{~ms}^{-1}\) (d) \(40 \mathrm{~ms}^{-1}\)

A body thrown vertically upward with an initial velocity \(u\) reaches maximum height in 6 second. The ratio of the distances travelled by the body in the first second and seventh second is (a) \(1: 1\) (b) \(11: 1 \quad\) (c) \(1: 2\) (d) \(1: 11\)

A parachutist after alling out falls \(50 \mathrm{~m}\) without friction. When parachute opens, it decelerates at \(2 \mathrm{~ms}^{-2}\). He reaches the ground with a speed of \(3 \mathrm{~ms}^{-1}\). At what height, did he fallen out? [AIEEE 2005] (a) \(111 \mathrm{~m}\) (b) \(293 \mathrm{~m}\) (c) \(182 \mathrm{~m}\) (d) \(91 \mathrm{~m}\)

A body projected vertically upwards crosses a point twice in it journey at a height \(h\) first after \(t_{1}\) and \(t_{2}\) second. Maximum height reached by the body is [EAMCET 2005] (a) \(2 g\left(t_{1}+t_{2}\right)\) (b) \(\frac{g}{4}\left(t_{1}+t_{2}\right)^{2}\) (c) \(\frac{g}{4}\left(t_{1} t_{2}\right)\) (d) \(2 g\left(\frac{t_{1}+t_{2}}{4}\right)^{2}\)

Water drops fall from a tap on the floor \(5 \mathrm{~m}\) below at regular intervals of time, the first drop striking the floor when the fifth drop begins to fall. The height at which the third drop will be, from ground, at the instant when first drop strikes the ground, will be \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(1.25 \mathrm{~m}\) (b) \(2.15 \mathrm{~m}\) (c) \(2.73 \mathrm{~m}\) (d) \(3.75 \mathrm{~m}\)

A ball is thrown vertically upwards from the top of a tower of height \(h\) with velocity \(v\). The ball strikes the ground after $$ \begin{array}{ll} \text { (a) } \frac{v}{g}\left[1+\sqrt{1+\frac{2 g h}{v^{2}}}\right] & \text { (b) } \frac{v}{g}\left[1+\sqrt{1-\frac{2 g h}{v^{2}}}\right] \\ \text { (c) } \frac{v}{g}\left(1+\frac{2 g h}{v^{2}}\right)^{1 / 2} & \text { (d) } \frac{v}{g}\left(1-\frac{2 g h}{v^{2}}\right)^{1 / 2} \end{array} $$

A body freely falling from rest has a velocity \(v\) after it falls through distance \(h .\) The distance it has to fall down further for its velocity to become double is (a) \(h\) (b) \(2 \mathrm{~h}\) (c) \(3 h\) (d) \(4 h\)

A particle starting from rest falls from a certain height. Assuming that the value of acceleration due to gravity remains the same throughout motion, its displacements in three successive half second intervals are \(S_{1}, S_{2}, S_{3} .\) Then, (a) \(S_{1}: S_{2}: S_{3}=1: 5: 9 \quad\) (b) \(S_{1}: S_{2}: S_{3}=1: 2: 3\) (c) \(S_{1}: S_{2}: S_{3}=1: 1: 1\) (d) \(S_{1}: S_{2}: S_{3}=1: 3: 5\)

A ball thrown upward from the top of a tower with speed \(v\) reaches the ground in \(t_{1}\) second. If this ball is thrown downward from the top of the same tower with speed \(v\) it reaches the ground in \(t_{2}\) second. In what time the ball shall reach the ground if it is allowed to falls freely under gravity from the top of the tower? \(\begin{array}{lll}\text { (a) } \frac{t_{1}+t_{2}}{2} & \text { (b) } \frac{t_{1}-t_{2}}{2} & \text { (c) } \sqrt{t_{1} t_{2}}\end{array}\) (d) \(t_{1}+t_{2}\)

A ball is dropped on the floor from a height of \(10 \mathrm{~m}\). It rebounds to a height of \(2.5 \mathrm{~m}\). If the ball is in contact with the floor for \(0.01 \mathrm{~s}\), the average acceleration during contact is nearly (Take \(g=10 \mathrm{~ms}^{-2}\) ) \(\begin{array}{ll}\text { (a) } 500 \sqrt{2} \mathrm{~ms}^{-2} \text { upwards } & \text { (b) } 1800 \mathrm{~ms}^{-2} \text { downwards }\end{array}\) \(\begin{array}{ll}\text { (c) } 1500 \sqrt{5} \mathrm{~ms}^{-2} \text { upwards } & \text { (d) } 1500 \sqrt{2} \mathrm{~ms}^{-2} \text { downwards }\end{array}\)

A stone thrown vertically upwards attains a maximum height of \(45 \mathrm{~m} .\) In what time the velocity of stone become equal to one-half the velocity of throw? (Given \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(2 \mathrm{~s}\) (b) \(1.5 \mathrm{~s}\) (c) \(1 \mathrm{~s}\) (d) \(0.5 \mathrm{~s}\)

A particle covers \(4 \mathrm{~m}, 5 \mathrm{~m}, 6 \mathrm{~m}\) and \(7 \mathrm{~m}\) in \(3 \mathrm{rd}, 4\) th, 5th and 6th second respectively. The particle starts (a) with an initial non-zero velcoity and moves with uniform acceleration (b) from rest and moves with uniform velocity (c) with an initial velocity and moves with uniform velcoity (d) from rest and moves with uniform acceleration

A balls is released from the top of a tower travels \(\frac{11}{36}\) of the height of the tower in the last second of its journey. The height of the tower is (Take \(g=10 \mathrm{~ms}^{-2}\) ) (a) \(11 \mathrm{~m}\) (b) \(36 \mathrm{~m}\) (c) \(47 \mathrm{~m}\) (d) \(180 \mathrm{~m}\)

At a metro station, a girl walks up a stationary escalator in time \(t_{1}\). If she remains stationary on the escalator, then the escalator take her up in time \(t_{2}\). The time taken by her to walk up on the moving escalator will be [NCERT Exemplar] (a) \(\left(t_{1}+t_{2}\right) / 2\) (b) \(t_{1} t_{2} /\left(t_{2}-t_{1}\right)\) (c) \(t_{1} t_{2} /\left(t_{2}+t_{1}\right)\) (d) \(t_{1}-t_{2}\)

A \(120 \mathrm{~m}\) long train is moving in a direction with speed \(20 \mathrm{~m} / \mathrm{s}\). A train \(B\) moving with \(30 \mathrm{~m} / \mathrm{s}\) in the opposite direction and \(130 \mathrm{~m}\) long crosses the first train in a time. (a) \(6 \mathrm{~s}\) (b) \(36 \mathrm{~s}\) (c) \(38 \mathrm{~s}\) (d) None of these

An express train is moving with a velocity \(v_{1}\) its driver finds another train is moving on the same track in the same direction with velocity \(v_{2}\). To avoid collision driver applies a retardation \(a\) on the train. The minimum time of avoiding collision will be (a) \(t=\frac{v_{1}-v_{2}}{a}\) (b) \(t=\frac{v_{1}^{2}-v_{2}^{2}}{2}\) (c) None (d) Both (a) and (b)

Rain drops fall vertically at a speed of \(20 \mathrm{~ms}^{-1}\). At what angle do they fall on the wind screen of a car moving with a velocity of \(15 \mathrm{~ms}^{-1}\), if the wind screen velocity inclined at an angle of \(23^{\circ}\) to the vertical? \(\left[\cot ^{-1}\left(\frac{4}{3}\right)=36^{\circ}\right]\) (a) \(60^{\circ}\) (b) \(30^{\circ}\) (c) \(45^{\circ}\) (d) \(90^{\circ}\)

A steam boat goes across a lake and comes back (i) on a quiet day when the water is still and (ii) on a rough day when there is a uniform current so as to help the journey onwards and to impede the journey back. If the speed of the launch on both days was same, the time required for complete journey on the rough day, as compared to the quiet day will be (a) more (b) less (c) same (d) None of these

Two trains travelling on the same track are approaching each other with equal speeds of \(40 \mathrm{~ms}^{-1}\). The drivers of the trains begin to decelerate simultaneously when they are just \(2 \mathrm{~km}\) apart. If the decelerations are both uniform and equal, then the value of deceleration to barely avoid collision should be (a) \(0.8 \mathrm{~ms}^{-2}\) (b) \(2.1 \mathrm{~ms}^{-2}\) (c) \(11.0 \mathrm{~ms}^{-2}\) (d) \(13.2 \mathrm{~ms}^{-2}\)

A \(210 \mathrm{~m}\) long train is moving due North at a speed of \(25 \mathrm{~m} / \mathrm{s}\). A small bird is flying due South, a little above the train with speed \(5 \mathrm{~m} / \mathrm{s}\). The time taken by the bird to cross the train is \(\begin{array}{llll}\text { (a) } 6 \mathrm{~s} & \text { (b) } 7 \mathrm{~s} & \text { (c) } 9 \mathrm{~s} & \text { (d) } 10 \mathrm{~s}\end{array}\)

A police jeep is chasing with velocity of \(45 \mathrm{~km} / \mathrm{h}\) a theif in another jeep moving with velocity \(153 \mathrm{~km} / \mathrm{h}\). Police fires a bullet with muzzle velocity of \(180 \mathrm{~m} / \mathrm{s}\). The velocity with which is will strike of the car of the thief is \(\begin{array}{lll}\text { (a) } 150 \mathrm{~m} / \mathrm{s} & \text { (b) } 27 \mathrm{~m} / \mathrm{s} & \text { (c) } 450 \mathrm{~m} / \mathrm{s}\end{array}\) (d) \(250 \mathrm{~m} / \mathrm{s}\)

A boat is sent across a river with a velocity of \(8 \mathrm{~km} / \mathrm{h}\). If the resultant velocity of boat is \(10 \mathrm{~km} / \mathrm{h}\), then velocity of river is \(\begin{array}{lll}\text { (a) } 10 \mathrm{~km} / \mathrm{h} & \text { (b) } 8 \mathrm{~km} / \mathrm{h} & \text { (c) } 6 \mathrm{~km} / \mathrm{h}\end{array}\) (d) \(4 \mathrm{~km} / \mathrm{h}\)

The distance between two particles moving towards each other is decreasing at the rate of \(6 \mathrm{~m} / \mathrm{s}\). If these particles travel with same speed and in the same direction then the separatioon increase at the rate of \(4 \mathrm{~m} / \mathrm{s}\). The particles have speed as \(\begin{array}{ll}\text { (a) } 5 \mathrm{~m} / \mathrm{s} 1 \mathrm{~m} / \mathrm{s} & \text { (b) } 4 \mathrm{~m} / \mathrm{s} ; 1 \mathrm{~m} / \mathrm{s}\end{array}\) (c) \(4 \mathrm{~m} / \mathrm{s} ; 2 \mathrm{~m} / \mathrm{s}\) (d) \(5 \mathrm{~m} / \mathrm{s} ; 2 \mathrm{~m} / \mathrm{s}\)

A train is moving towards east and a car is along north, both with same speed. The observed direction of a car to the passenger in the train is (a) east-north direction (b) west-north direction (c) south-east direction (d) None of the above

Two cars \(A\) and \(B\) are moving with same speed of \(45 \mathrm{~km} / \mathrm{h}\) along same direction. If a third car \(\mathrm{C}\) coming from the opposite direction with a speed of \(36 \mathrm{~km} / \mathrm{h}\) meets two cars in an interval of 5 minutes. The distance between cars \(A\) and \(B\) should be (in \(\mathrm{km}\) ) (a) \(6.75\) (b) \(7.25\) (c) \(5.55\) (d) \(8.35\)

Two trains \(A\) and \(B\) of length \(400 \mathrm{~m}\) each are moving on two parallel tracks with a uniform speed of \(72 \mathrm{~km} / \mathrm{h}\) in the same direction, with \(A\) ahead of \(B\). The driver of \(B\) decides to overtake \(A\) and accelerates by \(1 \mathrm{~m} / \mathrm{s}^{2}\). If after \(50 \mathrm{~s}\), the guard of \(B\) just brushes past the driver of \(A\), what was the original distance between them? (a) \(1250 \mathrm{~m}\) (b) \(1350 \mathrm{~m}\) (c) \(1450 \mathrm{~m}\) (d) None of these

On a two lane road, car \(A\) is travelling with a speed of \(36 \mathrm{~km} / \mathrm{h}\). Two cars \(B\) and \(C\) approach car \(A\) in opposite directions with a speed of \(54 \mathrm{~km} / \mathrm{h}\) each. At a certain instant, when the distance \(A B\) is equal to \(A C\), both being \(1 \mathrm{~km}, B\) decides to overtake \(A\) before \(C\) does. In this case, the acceleration of car \(B\) is required to avoid an accident (a) \(1 \mathrm{~m} / \mathrm{s}^{2}\) (b) \(0.1 \mathrm{~m} / \mathrm{s}^{2}\) (c) \(1.9 \mathrm{~m} / \mathrm{s}^{2}\) (d) \(0.2 \mathrm{~m} / \mathrm{s}^{2}\)

A passenger arriving in a new town wishes to go from the station to a hotel located \(10 \mathrm{~km}\) away on a straight road from the station. A dishonest cabman takes him along a circuitoius path \(23 \mathrm{~km}\) long and reaches the hotel in \(28 \mathrm{~min}\). What are the average speed of the taxi and the magnitude of average velocity respectively (in \(\mathrm{km} / \mathrm{h}) ?\) (a) \(49.3\) and \(21.43\) (b) \(48.3\) and \(24.43\) (c) 21 and 20 (d) \(21.3\) and \(49.3\)

A man can swim with a speed of \(4 \mathrm{~km} / \mathrm{h}\) in still water. How long does he take to cross a river \(1 \mathrm{~km}\) wide, if the river flows steadily \(3 \mathrm{~km} / \mathrm{h}\) and he makes his strokes normal to the river current. How far down the river does he go when he reaches the other bank? [NCERT] (a) \(850 \mathrm{~m}\) (b) \(750 \mathrm{~m}\) (c) \(650 \mathrm{~m}\) (d) None of these