Chapter 4

The horizontal range of a projectile fired at an angle of \(15^{\circ}\) is \(50 \mathrm{~m}\). If it is fired with the same speed at an angle of \(45^{\circ}\), its range will be (a) \(60 \mathrm{~m}\) (b) \(\underline{71} \bar{m}\) (c) \(100 \mathrm{~m}\) (d) \(141 \mathrm{~m}\)

A stone is projected with a velocity \(20 \sqrt{2} \mathrm{~ms}^{-1}\) at an angle of \(45^{\circ}\) to the horizontal. The average velocity of stone during its motion from starting point to its maximum height is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(5 \sqrt{5} \mathrm{~ms}^{-1}\) (b) \(10 \sqrt{5} \mathrm{~ms}^{-1}\) (c) \(20 \mathrm{~ms}^{-1}\) (d) \(20 \sqrt{5} \mathrm{~ms}^{-1}\)

A tennis ball rolls off the top of a sister case way with a horizontal velocity \(u \mathrm{~ms}^{-1}\). If the steps are \(b\) metre wide and \(h\) meter high, the ball will hit the edge of the \(n\)th step, if (al \(n=\frac{2 h u}{g b^{2}}\) (b) \(n=\frac{2 h u^{2}}{g b^{2}}\) (c) \(n=\frac{2 h u^{2}}{g b}\) (d) \(n=\frac{h u^{2}}{g b^{2}}\)

A bomber plane moves horizontally with a speed of \(500 \mathrm{~ms}^{-1}\) and a bomb releases from it, strikes the ground in 10 s. Angle at which it strikes the ground will be \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(\tan ^{-1}\left(\frac{1}{5}\right)\) (b) \(\tan \left(\frac{1}{5}\right)\) (c) \(\tan ^{-1}(1)\) (d) \(\tan ^{-1}(5)\)

A shell is fired from a cannon with a velocity \(v\) at angle \(\theta\) with horizontal. At the highest point, it explodes into two pieces of equal mass. One of the pieces retraces its path to the cannon. The speed of the other piece just after explosion is (a) \(3 v \cos \theta\) (b) \(2 v \cos \theta\) (c) \(\frac{3}{2} v \cos \theta\) (d) \(\frac{\sqrt{3}}{2} v \cos \theta\)

An aeroplane is flying in a horizontal direction with a velocity \(600 \mathrm{kmh}^{-1}\) at a height of \(1960 \mathrm{~m}\). When it is vertically above the point \(A\) on the ground, a body is dropped from it. The body strikes the ground at point \(B\). Calculate the distance \(A B\). (a) \(3.33 \mathrm{~km}\) (b) \(333 \mathrm{~km}\) (c) \(33.3 \mathrm{~km}\) (d) \(3330 \mathrm{~km}\)

The speed of projection of a projectile is increased by \(10 \%\), without changing the angle of projection. The percentage increase in the range will be (a) 109 (b) \(20 \%\) (c) \(15 \%\) (d) \(5 \%\)

The height \(y\) and distancex along the horizontal for a body projected in the xy-plane are given by \(y=8 t-5 t^{2}\) and \(x=6 t .\) The initial speed of projection is (a) \(8 \mathrm{~m} / \mathrm{s}\) (b) \(9 \mathrm{~m} / \mathrm{s}\) (c) \(10 \mathrm{~m} / \mathrm{s}\) (d) \((10 / 3) \mathrm{m} / \mathrm{s}\)

A projectile is launched with a speed of \(10 \mathrm{~m} / \mathrm{s}\) at an angle \(60^{\circ}\) with the horizontal from a sloping surface of inclination \(30^{\circ} .\) The range \(R\) is. (Take, \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (a) \(4.9 \mathrm{~m}\) (b) \(13.3 \mathrm{~m}\) (c) \(9.1 \mathrm{~m}\) (d) \(12.6 \mathrm{~m}\)

Two stones are projected with the same velocity in magnitude but making different angles with the horizontal. Their ranges are equal. If the angle of projection of one is \(\frac{\pi}{3}\) and its maximum height is \(y_{1}\), the maximum height of the other will be (a) \(3 y_{1}\) (b) \(2 y_{1}\) (c) \(\frac{y_{1}}{2}\) (d) \(\frac{y_{1}}{3}\)

A stone is just released from the window of a moving train along a horizontal straight track. The stone will hit the ground following (a) straight path (b) circular path (c) parabolic path (d) hyperbolic path

A car is travelling at a velocity of \(10 \mathrm{kmh}^{-1}\) on a straight road. The driver of the car throws a parcel with a velocity of \(10 \sqrt{2} \mathrm{kmh}^{-1}\) when the car is passing by a man standing on the side of the road. If the parcel is to reach the man, the direction of throw makes the following angle with direction of the car (a) \(135^{\circ}\) (b) \(45^{\circ}\) (c) \(\tan ^{-1}\left(\sqrt{2)} 60^{\circ}\right.\) (d) \(\tan \left(\frac{1}{\sqrt{2}}\right)\)

A particle is dropped from a height \(h .\) Another particle was what initially at a horizontal distance \(d\) from the first, is simultaneously projected with a horizontal velocity \(u\) and two particles just collide on the ground. The three quantities \(h, d\) and \(u\) are related to (a) \(d^{2}=\frac{u^{2} h}{2 g}\) (b) \(d^{2}=\frac{2 u^{2} h}{g}\) (c) \(d=h\) (d) \(g d^{2}=u^{2} h\)

A particle moves in the \(x y\)-plane with velocity \(v_{x}=8 t-2\) and \(v_{y}=2\) If it passes through the point \(x=14\) and \(y=4\) at \(t=2\) s, find the equation \((x-y\) relation) of the path. (a) \(x=y^{2}-y+2\) (b) \(x=2 y^{2}+2 y-3\) (c) \(x=3 y^{2}+5\) (d) Cannot be found from above data

A body of mass \(1 \mathrm{~kg}\) is projected with velocity \(50 \mathrm{~m} / \mathrm{s}\) at an angle of \(30^{\circ}\) with the horizontal. At the highest point of its path a force \(10 \mathrm{~N}\) starts acting on body for 5 s vertically upward besides gravitational force, what is horizontal range of the body? ( \(g=10 \mathrm{~m} / \mathrm{s}^{2}\) ) (a) \(125 \sqrt{3} \mathrm{~m}\) (b) \(200 \sqrt{3} \mathrm{~m}\) (c) \(500 \mathrm{~m}\) (d) \(250 \sqrt{3} \mathrm{~m}\)

The ceiling of a long hall is \(25 \mathrm{~m}\) high. Then, the maximum horizontal distance that a ball thrown with a speed of \(40 \mathrm{~m} / \mathrm{s}\) can go without hitting the ceiling of the hall, is [NCERT Exemplar] (a) \(95.5 \mathrm{~m}\) (b) \(105.5 \mathrm{~m}\) (c) \(100 \mathrm{~m}\) (d) \(150.5 \mathrm{~m}\)

If a stone is to hit at a point which is at a distance \(d\) away and at a height \(h\) above the point from where the stone starts, then what is the value of initial speed \(u\), if the stone is launched at an angle \(\theta\) ? (a) \(\frac{g}{\cos \theta} \sqrt{\frac{d}{2(d \tan \theta-h)}}\) (b) \(\frac{d}{\cos \theta} \sqrt{\frac{d}{2(d \tan \theta-h)}}\) (c) \(\sqrt{\frac{g d^{2}}{h \cos ^{2} \theta}}\) (d) \(\sqrt{\frac{g d^{2}}{(d-h)}}\)

A particle leaves the origin with an initial velocity \(\mathbf{v}=(3.00 \hat{\mathrm{i}}) \mathrm{ms}^{-1}\) and a constant acoeleration \(\mathbf{a}=(-1.00 \hat{\mathrm{i}}-0.50 \hat{\mathrm{j}}) \mathrm{ms}^{-2}\). When the particle reaches its maximum \(x\)-coordinate, what is its \(y\)-component a velocity? (a) \(-2.0 \mathrm{~ms}^{-1}\) (b) \(-1.0 \mathrm{~ms}^{-1}\) (c) \(-1.5 \mathrm{~ms}^{-1}\) (d) \(1.0 \mathrm{~ms}^{-1}\)

After one second the velocity of a projectile makes an angle of \(45^{\circ}\) with the horizontal. After another one second it is travelling horizontally. The magnitude of its initial velocity and angle of projection are \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(14.62 \mathrm{~ms}^{-1}, \tan ^{-1}(2)\) (b) \(22.36 \mathrm{~ms}^{-1}, \tan ^{-1}(2)\) (c) \(14.62 \mathrm{~ms}^{-1}, 60^{\circ}\) (d) \(22.36 \mathrm{~ms}^{-1}, 60^{\circ}\)

A bomb is dropped on an enemy post by an aeroplane flying horizontally with a velocity of \(60 \mathrm{kmh}^{-1}\) and at a height of \(490 \mathrm{~m}\). At the time of dropping the bomb, how far the aeroplane should be from the enemy post so that the bomb may directly hit the target? (a) \(\frac{400}{3} \mathrm{~m}\) (b) \(\frac{500}{3} \mathrm{~m}\) (c) \(\frac{1700}{3} \mathrm{~m}\) (d) \(498 \mathrm{~m}\)

A particle of mass \(m\) is projected with a velocity \(v\) at an angle of \(60^{\circ}\) with horizontal. When the particle is at its maximum height. The magnitude of its angular momentum about the point of projection is (a) zero (b) \(\frac{3 \mathrm{mv}^{2}}{16 \mathrm{~g}}\) (c) \(\frac{\sqrt{3} m v^{2}}{16 g}\) (d) \(\frac{3 m v^{2}}{3 g}\)

A body projected with velocity \(u\) at projection angle \(\theta\) has horizontal range \(R\). For the same velocity and projection angle, its range on the moon surface will be \(g_{\text {moen }}=g_{\text {earth }} / 6\) ) (a) \(36 R\) (b) \(\frac{R}{36}\) (c) \(\frac{R}{16}\) (d) \(6 R\)

A boy throws a ball with a velocity \(u\) at an angle \(\theta\) with the horizontal. At the same instant he starts running with uniform velocity to eatch the ball before if hits the ground. To achieve this he should run with a velocity of (a) \(u \cos \theta\) (b) \(u \sin \theta\) (c) \(u \tan \theta\) (d) \(u \sec \theta\)

A body of mass \(m\) thrown horizontally with velocity \(v\), from the top of tower of height \(h\) touches the level ground at distance of \(250 \mathrm{~m}\) from the foot of the tower. A body of mass \(2 \mathrm{~m}\) thrown horizontally with velocity \(\frac{v}{2}\), from the top of tower of height \(4 h\) will touch the level ground at a distance \(x\) from the foot of tower. The value of \(x\) is (a) \(250 \mathrm{~m}\) (b) \(500 \mathrm{~m}\) (c) \(125 \mathrm{~m}\) (d) \(250 \sqrt{2} \mathrm{~m}\)

Two projectiles \(A\) and \(B\) thrown with speeds in the ratio \(1: \sqrt{2}\) acquired the same heights. If \(A\) is thrown at an angle of \(45^{\circ}\) with the horizontal, the angle of projection of \(B\) will be (a) \(0^{\circ}\) (b) \(60^{\circ}\) (c) \(30^{\circ}\) (d) \(45^{\circ}\) (e) \(15^{\circ}\)

A particle is projected with a velocity \(200 \mathrm{~ms}^{-1}\) at an angle of \(60^{\circ} .\) At the highest point, it explodes into three particles of equal masses. One goes vertically upwards with a velocity \(100 \mathrm{~ms}^{-1}\), the second particle goes vertically downwards. What is the velocity of third particle? (a) \(120 \mathrm{~ms}^{-1}\) making \(60^{\circ}\) angle with horizontal (b) \(200 \mathrm{~ms}^{-1}\) making \(30^{\prime}\) angle with horizontal (c) \(300 \mathrm{~ms}^{-1}\) (d) \(200 \mathrm{~ms}^{-1}\)

Two paper sereen \(A\) and \(B\) are separated by a distance of \(100 \mathrm{~m}\). A bullet pierces \(A\) and \(B\). The hole in \(B\) is \(10 \mathrm{~cm}\) below the hole in \(A\). If the bullet is travelling horizontally at the time of hitting \(A\). Then, the velocity of the bullet at \(A\) is (a) \(100 \mathrm{~m} / \mathrm{s}\) (b) \(200 \mathrm{~m} / \mathrm{s}\) (c) \(600 \mathrm{~m} / \mathrm{s}\) (d) \(700 \mathrm{~m} / \mathrm{s}\)

The trajectory of a projectile in vertical plane is \(y=a x-b x^{2}\), where \(a\) and \(b\) are constants and \(x\) and \(y\) are respectively horizontal and vertical distances of the projectile from the point of projection. The maximum height attained by the particle and the angle of projection from the horizontal are (a) \(\frac{b^{2}}{4 b}, \tan ^{-1}(b)\) (b) \(\frac{a^{2}}{b}, \tan ^{-1}(2 b)\) (c) \(\frac{a^{2}}{4 b}, \tan ^{-1}(a)\) (d) \(\frac{2 a^{2}}{b}, \tan ^{-1}(a)\)

Two projectiles thrown from the same point at angles \(60^{\circ}\) and \(30^{\circ}\) with the horizontal attain the same height. The ratio of their initial velocities is (a) 1 (b) 2 (c) \(\sqrt{3}\) (d) \(\frac{1}{\sqrt{3}}\)

Two projectiles \(A\) and \(B\) are projected with same speed at angles \(15^{\circ}\) and \(75^{\circ}\) respectively to the maximum and have same horizontal range. If \(h\) be the maximum height and \(T\) total time of flight of a projectile, then (a) \(h_{A}>h_{B}\) (b) \(h_{A}T_{B}\)

Two particles are projected in air with speed \(v_{0}\) at angles \(\theta_{1}\) and \(\theta_{2}\) (both acute) to the horizontal, respectively. If the height reached by the first particle is greater than that of the second, then tick the right choices (a) angle of projection : \(\theta_{1}>\theta_{2}\) (b) time of flight : \(T_{1}>T_{2}\) (c) horizontal range : \(R_{1}>R_{2}\) (d) total energy : \(U_{1}>U_{2}\)

A projectile has the same range \(R\) for two angles of projections. If \(T_{1}\) and \(T_{2}\) be the times of flight in the two cases, then (using \(\theta\) as the angle of projection corresponding to \(T_{1}\) ) (a) \(T_{1} T_{2} \propto R\) (b) \(T_{1} T_{2} \propto R^{2}\) (c) \(T_{1} / T_{2}=\tan \theta\) (d) \(T_{1} / T_{2}=1\)

A particle is projected with a velocity of \(30 \mathrm{~m} / \mathrm{s}\), at an angle \(\theta_{0}=\tan ^{-1}\left(\frac{3}{4}\right)\). After \(1 \mathrm{~s}\), the particle is moving at an angle \(\theta\) to the horizontal, where \(\tan \theta\) will be equal to \(\left(g=10 \mathrm{~m} / \mathrm{s}^{2}\right)\) (a) \(]\) (b) 2 (c) \(\frac{1}{2}\) (d) \(\frac{1}{3}\)

When a projectile is projected at a certain angle with the horizontal, its horizontal range is \(R\) and time of flight is \(T_{1}\). When the same projectile is throwing with the same speed at some other angle with the horizontal, its horizontal range is \(R\) and time of flight is \(T_{2}\). The produet of \(T_{1}\) and \(T_{2}\) is (a) \(\frac{R}{g}\) (b) \(\frac{2 R}{g}\) (c) \(\frac{3 R}{g}\) (d) \(\frac{4 R}{g}\)

Two stones thrown at different angles have same initial velocity and same range. If \(H\) is the maximum height attained by one stone thrown at an angle of \(30^{\circ}\), then the maximum height attained by the other stone is (a) \(\frac{H}{2}\) (b) \(\underline{H}\) (c) \(2 H\) (d) \(3 \mathrm{H}\)

The maximum height attained by projectile is (a) \(2 \mathrm{~h} / 3\) (b) \(3 h\) (c) \(3 h / 4\) (d) \(3 h / 2\) Two second after projection, a projectile is travelling in a direction inclined at \(30^{\circ}\) to the horizontal. After I more second, it is travelling horizontally (use \(g=10 \mathrm{~ms}^{-2}\) )

A projectile shot into air at some angle with the horizontal has a range of \(200 \mathrm{~m}\). If the time of flight is 5 s, then the horizontal component of the velocity of the projectile at the highest point of trajectory is (a) \(40 \mathrm{~ms}^{-1}\) (b) \(0 \mathrm{~ms}^{-1}\) (c) \(9.8 \mathrm{~ms}^{-1}\) (d) equal to the velocity of projection of the projectile

The kinetic energy of a project at the height point is half of the initial kinetic energy. What is the angle of projection with the horizontal? [a) \(3 \overline{0^{\circ}}\) (b) \(45^{*}\) (c) \(6 \overline{0^{n}}\) (d) \(90^{*}\)

A ball is projected from a certain point on the surface of a planet at a certain angle with the horizontal surface. The horizontal and vertical displacement \(x\) and \(y\) vary with time \(t\) in second as \(x=10 \sqrt{3} t\) and \(y=10 t-t^{2}\). The maximum height attained by the ball is (a) \(100 \mathrm{~m}\) (b) \(75 \mathrm{~m}\) (c) \(50 \mathrm{~m}\) (d) \(25 \mathrm{~m}\)

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choices, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given ahead (a) If both Assertion and Reason are true and 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 If a particle is projected vertices upwards with velocity \(u\), the maximum height attained by the particle is \(h_{1}\). The same particle is projected at angle \(30^{\circ}\) from horizontal with the same speed \(u\). Now the maximum height is \(h_{2}\). Thus, \(h_{1}=4 h_{2}\) Reason In first case \(v=0\) at highest point and in second case \(v \neq 0\) at highest point.

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choices, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given ahead (a) If both Assertion and Reason are true and 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 At highest point of a projectile dot product of velocity and acceleration is zero. Reason At highest point velocity and acceleration are mutually perpendicular.

For a projectile thrown into space with a speed \(v\), the horizontal range is \(\frac{\sqrt{3} v^{2}}{2 g} .\) The vertical range is \(\frac{v^{2}}{8 g}\). The angle which the projectile makes with the horizontal initially is (a) \(15^{*}\) (b) \(30^{*}\) (c) \(45^{\circ}\) [d) \(60^{\circ}\)

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choices, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given ahead (a) If both Assertion and Reason are true and 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 particle is projected with speed \(u\) at an angle \(\theta\) with the horizontal. At any time during motion, speed of particle is \(v\) at angle \(\alpha\) with the vertical, then \(v \sin \alpha\) is always constant throughout the motion. Reason In case of projectile motion, magnitude of radial acceleration at top most point is maximum.

The velocity of projection of an oblique projectile is \((6 \hat{\mathbf{i}}+\mathbf{8} \hat{\mathbf{j}}) \mathrm{ms}^{-1}\). The horizontal range of the projectile is (a) \(4.9 \mathrm{~m}\) (b) \(9.6 \mathrm{~m}\) (c) \(19.6 \mathrm{~m}\) (d) \(14 \mathrm{~m}\)

Assertion-Reason type. Each of these contains two Statements: Statement I (Assertion), Statement II (Reason). Each of these questions also has four alternative choices, only one of which is correct. You have to select the correct choices from the codes (a), (b), (c) and (d) given ahead (a) If both Assertion and Reason are true and 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 If in a projectile motion, we take air friction into consideration, then \(t_{\text {ascent }}

Two projectiles \(A\) and \(B\) are thrown with velocities \(v\) and \(\frac{v}{2}\) respectively. They have the same range. If \(B\) is thrown at an angle of \(15^{\circ}\) to the horizontal, A must have been thrown at an angle (a) \(\sin ^{-1}\left(\frac{1}{16}\right)\) (b) \(\sin ^{-1}\left(\frac{1}{4}\right)\) (c) \(2 \sin ^{-1}\left(\frac{1}{4}\right)\) (d) \(\frac{1}{2} \sin ^{-1}\left(\frac{1}{8}\right)\)

A projectile is given an initial velocity of \((\hat{\mathbf{i}}+2 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}\), where \(\hat{\mathbf{i}}\) is along the ground and \(\hat{\mathbf{j}}\) is along the vertical. If \(g=10 \mathrm{~m} / \mathrm{s}^{2}\), the equation of its trajectory is [JEE Main 2013] (a) \(y=x-5 x^{2}\) (b) \(y=2 x-5 x^{2}\) (c) \(4 y=2 x-5 x^{2}\) (d) \(4 y=2 x-25 x^{2}\)

Two cars of masses \(m_{1}\) and \(m_{2}\) are moving in circles of radii \(r_{1}\) and \(r_{2}\) respectively. Their speeds are such that they make complete circles in the same time \(t\). The ratio of their centripetal acceleration is [AIEEE 2012] (a) \(m_{1} r_{1}: m_{2} 5\) (b) \(m_{1}: m_{2}\) (c) \(r_{1}: r_{2}\) (d) \(1: 1\)

The horizontal range of an oblique projectile is equal to the distance through which a projectile has to fall freely from rest to acquire a velocity equal to the velocity of projection in magnitude. The angle of projection is (a) \(15^{\circ}\) (b) \(60^{*}\) (c) \(45^{\circ}\) (d) \(3 \overline{0^{*}}\)

A particle of mass \(m\) is projected with a velocity \(v\) making an angle of \(30^{\circ}\) with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height \(h\) is (a) \(\frac{\sqrt{3}}{2} \frac{m v^{2}}{g}\) (b) zero (c) \(\frac{m v^{3}}{\sqrt{2} g}\) (d) \(\frac{\sqrt{3}}{16} \frac{m v^{3}}{g}\)