Chapter 4

Two stones are projected so as to reach the same distance from the point of projection on a horizontal surface. The maximum height reached by one exceeds the other by an amount equal to half the sum of the height attained by them. Then, angle of projection of the stone which attains smaller height is [a] \(45^{\circ}\) (b) \(60^{\circ}\) (c) \(3 \overline{0^{*}}\) (d) \(\tan ^{-1}(3 / 4)\)

A large number of bullets are fired in the all directions with same speed \(v\). What is the maximum area on the ground on which these bullets will spread? (a) \(\pi \frac{v^{2}}{g}\) (b) \(\pi \frac{v^{4}}{g^{2}}\) (c) \(n^{2} \frac{v^{4}}{g^{2}}\) (d) \(\pi^{2} \frac{v^{2}}{g^{2}}\)

A ball rolls of the top of stair-way with a horizontal velocity of magnitude \(1.8 \mathrm{~ms}^{-1}\). The steps are \(0.20 \mathrm{~m}\) high and \(0.20 \mathrm{~m}\) wide. Which step will the ball hit first? (a) First (b) Second (c) Third (d) Fourth

A projectile is projected with velocity \(k v_{e}\) vertically upward direction from the ground into the space \(\left(v_{e}\right.\) is the escape velocity and \(\left.k<1\right)\). If air resistance is considered to be negligible then the maximum height from the centre of earth to which it can go will be \((R=\) radius of earth \() \quad\) [Orissa JEE 2010] (a) \(\frac{R}{k^{2}+1}\) (b) \(\frac{R}{k^{2}-1}\) (c) \(\frac{R}{1-k^{2}}\) (d) \(\frac{R}{k+1}\)

A ball is projected up an incline of \(30^{\circ}\) with a velocity of \(30 \mathrm{~ms}^{-1}\) at an angle of \(30^{\circ}\) with reference to the inclined plane from the bottom of the inclined plane. If \(g=10 \mathrm{~ms}^{-2}\), then the range on the inclined plane is (a) \(12 \mathrm{~m}\) (b) \(60 \mathrm{~m}\) (c) \(120 \mathrm{~m}\) (d) \(600 \mathrm{~m}\)

A point \(P\) moves in counter-clockwise direction on a circular path as shown in figure. The movement of \(P\) is such that it sweep out a length \(s=t^{3}+5\), where \(s\) is in metres and \(t\) is in seconds. The radius of the path is \(20 \mathrm{~m}\). The acceleration of \(P\) when \(t=2 \mathrm{~s}\) is nearly [AIEEE 2010] (a) \(14 \mathrm{~m} / \mathrm{s}^{2}\) (b) \(13 \mathrm{~m} / \mathrm{s}^{2}\) (c) \(12 \mathrm{~m} / \mathrm{s}^{2}\) (d) \(7.2 \mathrm{~m} / \mathrm{s}^{2}\)

The maximum range of projectile fired with some initial velocity is found to be \(1000 \mathrm{~m}\), in the absence of wind and air resistance. The maximum height reached by the projectile is [Orissa JEE 2009] (a) \(250 \mathrm{~m}\) (b) \(500 \mathrm{~m}\) (c) \(1000 \mathrm{~m}\) (d) \(2000 \mathrm{~m}\)

A piece of marble is projected from earth's surface with velocity of \(50 \mathrm{~ms}^{-1}, 2 \mathrm{~s}\) later it just clears a wall \(5 \mathrm{~m}\) high. What is the angle of projection? (a) \(45^{\circ}\) (b) \(30^{*}\) (c) \(60^{\circ}\) (d) None of these

A particle is projected with velocity \(v_{0}\) along \(x\)-axis. The deceleration on the particle is proportional to the square of the distance from the origin, ie., \(a=\alpha x^{2}\), the distance at which the particle stop is (a) \(\sqrt{\frac{3 v_{0}}{2 \alpha}}\) (b) \(\left(\frac{3 v_{0}}{2 \alpha}\right)^{1 / 3}\) (c) \(\sqrt{\frac{2 v_{0}^{2}}{3 \alpha}}\) (d) \(\left(\frac{3 v_{0}^{2}}{2 \alpha}\right)^{1 / 3}\)

A body is projected with speed \(v \mathrm{~ms}^{-1}\) at angle \(\theta\). The kinetic energy at the highest point is half of the initial kinetic energy. The value of \(\theta\) is (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(90^{\circ}\)

If a body is projected with an angle to the horizontal, then \(\quad\) [EAMCET 2008] (a) its velocity is always perpendicular to its acceleration (b) its velocity becomes zero at maximum height (c) its velocity makes zero angle with the horizontal at its maximum height (d) the body just before hitting the ground, the direction of velocity coincides with the acceleration

A body is thrown upwards from the earth surface with velocity \(5 \mathrm{~ms}^{-1}\) and from a planet surface with velocity \(3 \mathrm{~ms}^{-1}\). Both follow the same path. What is the projectile acceleration due to gravity on the planet? Acceleration due to gravity on earth is \(10 \mathrm{~ms}^{-1} .\) [Orissa JEE 2008] (a) \(2 \mathrm{~ms}^{-2}\) (b) \(3.6 \mathrm{~ms}^{-2}\) (c) \(4 \mathrm{~ms}^{-2}\) (d) \(5 \mathrm{~ms}^{-2}\)

The angle of projection of a projectile for which the horizontal range and maximum height are equal to (a) \(\tan ^{-1}(2)\) (b) \(\tan ^{-1}(4)\) (c) \(\cot ^{-1}(2)\) (d) \(60^{\circ}\)

Two particles \(A\) and \(B\) are projected with same speed so that the ratio of their maximum heights reached is \(3: 1\) If the speed of \(A\) is doubled without altering other parameters, the ratio of the horizontal ranges obtained by \(A\) and \(B\) is \(\quad\) [Kerala CET 2008] (a) \(1: 1\) (b) \(2: 1\) (c) \(4: 1\) (d) \(3: 2\)

Two bodies are projected from the same point with equal speeds in such directions that they both strike the same point on a plane whose inclination is \(\beta\). If \(\alpha\) be the angle of projection of the first body with the horizontal the ratio of their times of flight is (a) \(\frac{\cos \alpha}{\sin (\alpha+\beta)}\) (b) \(\frac{\sin (\alpha+\beta)}{\cos \alpha}\) (c) \(\frac{\cos \alpha}{\sin (\alpha-\beta)}\) (d) \(\frac{\sin (\alpha-\beta)}{\cos \alpha}\)

A particle is projected with certain velocity at two different angles of projections with respect to horizontal plane so as to have same range \(R\) on a horizontal plane. If \(t_{1}\) and \(t_{2}\) are the time taken for the two paths, the which one of the following relations is correct? [UP SEE 2008] (a) \(t_{1} t_{2}=\frac{2 R}{g}\) (b) \(t_{1} t_{2}=\frac{R}{g}\) (c) \(t_{1} t_{2}=\frac{g}{2 g}\) (d) \(t_{1} t_{2}=\frac{4 R}{g}\)

A particle is projected with velocity \(2 \sqrt{g h}\) so that it just clears two walls of equal height \(h\), which are at a distance of \(2 h\) from each other. What is the time interval of passing between the two walls? (a) \(\frac{2 h}{g}\) (b) \(\sqrt{\frac{g h}{g}}\) (c) \(\sqrt{\frac{h}{g}}\) (d) \(2 \sqrt{\frac{h}{g}}\)

A particle is projected at \(60^{\circ}\) to the horizontal with an energy \(E .\) The kinetic energy and potential energy at the highest point are [KCET, AIEEE 2007] (a) \(\left(\frac{E}{2}, \frac{E}{2}\right)\) (b) \(\left(\frac{3 E}{4}, \frac{E}{4}\right)\) (c) \((E, 0)\) (d) \(\left(\frac{E}{4}, \frac{3 E}{4}\right)\)

The maximum height attained by a projectile when thrown at an angle \(\theta\) with the horizontal is found to be half the horizontal range. Then, \(\theta\) is equal to [KCET 2007] (a) \(\tan ^{-1}(2)\) (b) \(\frac{\pi}{6}\) (c) \(\frac{\pi}{4}\) (d) \(\tan ^{-1}\left(\frac{1}{2}\right)\)

A particle is projected from the ground with an initial speed of \(v\) at an angle \(\theta\) with horizontal. The average velocity of the particle between its point of projection and highest point of trajectory is (a) \(\frac{v}{2} \sqrt{1+2 \cos ^{2} \theta}\) (b) \(\frac{v}{2} \sqrt{1+\cos ^{2} \theta}\) (c) \(\frac{v}{2} \sqrt{1+3 \cos ^{2} \theta}\) (d) \(v \cos \theta\)

A particle is thrown in the upward direction making an angle of \(60^{\circ}\) with the horizontal direction with a velocity of \(147 \mathrm{~ms}^{-1}\). Then, the time after which its inclination with the horizontal is \(45^{\circ}\), is IUP SEE 2006] (a) \(15 \mathrm{~s}\) (b) \(10.98 \mathrm{~s}\) (c) \(5.49 \mathrm{~s}\) (d) \(2.745 \mathrm{~s}\)

A body of mass \(m\) is thrown upward at an angle \(\theta\) with the horizontal with velocity \(v\). While rising up the velocity of the mass after \(t\) second will be (a) \(\sqrt{[v \cos \theta)^{2}+(v \cdot \sin \theta)^{2}}\) (b) \(\sqrt{(v \cos \theta-v \sin \theta)^{2}-g t}\) (c) \(\sqrt{v^{2}+g^{2} t^{2}-(2 v \sin \theta) g t}\) (d) \(\sqrt{v^{2}+g^{2} t^{2}-(2 v \cos \theta) g t}\)

The equation of motion of a projectile are given by \(x=36 t .\) If and \(2 y=96 t-9.8 t^{2} \mathrm{~m}\). The angle of projectile is (a) \(\sin ^{-1}\left(\frac{4}{5}\right)\) (b) \(\sin ^{-1}\left(\frac{3}{5}\right)\) (c) \(\sin ^{-1}\left(\frac{4}{3}\right)\) (d) \(\sin ^{-1}\left(\frac{3}{4}\right)\)

A fighter plane enters inside the enemy territory, at time \(t=0\) with velocity \(v_{0}=250 \mathrm{~ms}^{-1}\) and moves horizontally with constant acceleration \(a=20 \mathrm{~ms}^{-2}\) (see figure). An enemy tank at the border, spot the plane and fire shots at an angle \(\theta=60^{\circ}\) with the horizontal and with velocity \(u=600 \mathrm{~ms}^{-1}\). At what altitude \(H\) of the plane it can be hit by the shot? (a) \(1500 \sqrt{3} \mathrm{~m}\) (b) \(125 \mathrm{~m}\) (c) \(1400 \mathrm{~m}\) (d) \(2473 \mathrm{~m}\)

An aircraft, diving at an angle of \(53.0^{\circ}\) with the vertical releases a projectile at an altitude of \(730 \mathrm{~m}\). The projectile hits the ground \(5.00 \mathrm{~s}\) after being released. What is the speed of the aircraft? (a) \(282 \mathrm{~ms}^{-1}\) (b) \(202 \mathrm{~ms}^{-1}\) (c) \(182 \mathrm{~ms}^{-1}\) (d) \(102 \mathrm{~ms}^{-1}\)

A particle \(A\) is projected from the ground with an initial velocity of \(10 \mathrm{~ms}^{-1}\) at an angle of \(60^{\circ}\) with horizontal. From what height \(h\) should an another particle \(B\) be projected horizontal with veloeity \(5 \mathrm{~ms}^{-1}\) go that both the particles collide with velocity \(5 \mathrm{~ms}^{-1}\) so that both the particles collide on the ground at point \(C\) if both are projected simultaneously? \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(10 \mathrm{~m}\) (b) \(30 \mathrm{~m}\) [c) \(15 \mathrm{~m}\) (d) \(25 \mathrm{~m}\)

A very broad elevator is going up vertically with a constant acceleration of \(2 \mathrm{~ms}^{-2}\). At the instant when its velocity is \(4 \mathrm{~ms}^{-1}\) a ball is projected from the floor of the list with a speed of \(4 \mathrm{~ms}^{-1}\) relative to the floor at an elevation of \(30^{\circ}\). The time taken by the ball to return the floor is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(1 / 2 s\) (b) \(1 / 3 \mathrm{~s}\) (c) \(1 / 4 \mathrm{~s}\) (d) \(1 \mathrm{~s}\)

A projectile is fired at an angle of \(30^{\circ}\) to the horizontal such that the vertical component of its initial velocity is \(80 \mathrm{~ms}^{-1}\), Its time of flight is \(T\). Its velocity at \(t=\frac{T}{4}\) has a magnitude of nearly (a) \(200 \mathrm{~ms}^{-1}\) (b) \(300 \mathrm{~ms}^{-1}\) (c) \(140 \mathrm{~ms}^{-1}\) (d) \(100 \mathrm{~ms}^{-1}\)

A particle is projected from the ground at an angle of \(60^{\text {n }}\) with horizontal with speed \(u=20 \mathrm{~ms}^{-1}\). The radius of curvature of the path of the particle, when its velocity makes an angle of \(30^{\circ}\) with horizontal is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (a) \(10.6 \mathrm{~m}\) (b) \(12.8 \mathrm{~m}\) (c) \(15.4 \mathrm{~m}\) (d) \(24.2 \mathrm{~m}\)