COM in Collision and Explosion

Q. A particle of mass $m=4\text{kg}$ is projected with a speed of $u=10\text{m/s}$ at an angle of ${45}^{\circ }$ with the horizontal. The particle explodes in mid air when it is at the maximum height. One component weighing $1\text{kg}$ comes to rest after the explosion and the other component of mass $3\text{kg}$ moves away. Find the distance where the $3\text{kg}$ mass component strikes the ground from the initial position (projection point). [Take $g=10{\text{m/s}}^2$]

1. $15\text{m}$
2. $\frac{35}{3}\text{m}$
3. $\frac{20}{3}\text{m}$
4. $25\text{m}$

Q. A projectile of mass $2\text{m}$ explodes at the highest point of its path. It breaks into two equal parts. One part retraces its path. Find the distance of the second part from the point of projection when it finally lands on the ground. Given that range of the projectile would be $200\text{m}$ if no explosion had taken place.

1. $100\text{m}$
2. $400\text{m}$
3. $200\text{m}$
4. $150\text{m}$

Q.

This question has statement I and Statement II. Of the four choices given after the statements, choose the one that best describes the two statements. Statement I A point particle of mass m moving with speed v collides with stationary point particle of mass M. If the maximum energy loss possible is given as $f\left(\frac{1}{2}m{v}^2\right),thenf=(\frac{m}{M}+m)$ Assertion and Reason

1. If Statement I is true, Statement II it true; Statement II is the correct explanation for Statement I

2. If Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I

3. If Statement I is true; Statement II is false

4. If Statement I is false; Statement II is true

Q.

A projectile is fired at a speed of 100 m/s at an angle of ${37}^{\circ }$ above the horizontal. At the highest point, the projectile breaks into two parts of mass in ratio 1 : 3, the smaller coming to rest. Find the distance from the launching point to the point where the heavier piece lands.

1. 1120 m

2. 1220 m

3. 1020 m

4. 920 m

Q. Two blocks of masses $m_1$ and $m_2$ connected by a massless spring of spring constant 'k' rest on a smooth horizontal plane as shown in figure. Block 2 is shifted a small distance 'x' to the left and the released. Find the velocity of centre of mass of the system after block 1 breaks off the wall.

1. $\frac{\sqrt{m_{2}K}}{m_1+m_2}$x
2. $\frac{\sqrt{m_{1}K}}{m_1+m_2}$x
3. $\frac{\sqrt{(m_1+m_2)K}}{m_1+m_2}x$
4. zero

Q.

A particle of mass m moving in the x-direction with speed 2v is hit by another particle of mass 2m moving in the y-direction with speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to

1. 50%

2. 56%

3. 62%

4. 44%

Q. If the kinetic energy of the particle is increased to $16$ times its previous value. The percentage change in the de-Broglie wavelength of the particle is

1. $25$
2. $60$
3. $50$
4. $75$

Q. A particle is projected with a speed $u$ at an angle $\theta$ with horizontal from point $A$. It strikes elastically with a vertical wall at height $h/2$. It rebounds and reaches maximum height $h$ and falls back on the ground at point $B$ as shown in Figure. Distances from $A$ to wall and from wall to $B$ are $x_1$ and $x_2$, and time to cover is $t_1$ and $t_2$ respectively. Match the values in column I with the expressions in column II.


$$\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{i.}\sqrt{2}& \text{a.}\frac{x_2-x_1}{x_2+x_1}\text{or}\frac{x_2+x_1}{x_2-x_1}\\ \text{ii.}\frac{1}{\sqrt{2}}& \text{b.}\frac{t_2-t_1}{t_2+t_1}\text{or}\frac{t_2+t_1}{t_2-t_1}\\ \text{iii.}1& \text{c.}\frac{u\sin \theta }{g(t_2+t_1)}\\ \text{iv.}\frac{1}{2}& \text{d.}\frac{u\cos \theta (t_1+t_2)}{x_1+x_2}\end{array}$$

1. $\text{(i) a, b (ii) a, b (iii) d (iv) c}$
2. $\text{(i) a (ii) b (iii) d (iv) c}$
3. $\text{(i) a, b (ii) a, b (iii) d (iv) c}$
4. $\text{(i) a (ii) a, c (iii) d (iv) b}$

Q. A particle of mass $4\text{m}$ is projected from the ground at some angle with horizontal. Its horizontal range is $R$. At the highest point of its path it breaks into two pieces of masses $m$ and $3\text{m}$ respectively such that the smaller mass comes to rest. The larger mass finally falls at a distance $x$ from the point of projection, where $x$ is equal to

1. $\frac{2R}{3}$
2. $\frac{7R}{6}$
3. $\frac{5R}{4}$
4. None of these.

Q.

A ball of mass 'm', moving with a velocity 'v' along X-axis, strikes another ball of mass '2m' kept at rest. The first ball comes to rest after collision and the other breaks into two equal pieces. One of the pieces starts moving along Y-axis with a speed $v_1$. What will be the velocity of the other piece?

1. $\sqrt{v^2+v_1^2}$

2. $\sqrt{2v^2+v_1^2}$

3. $\sqrt{v^2-v_1^2}$

4. $\sqrt{\sqrt{v^2+2v_1^2}}$

Q. A bomb is projected with initial velocity $9.8\text{m/s}$ making ${15}^{\circ }$ angle with the horizontal. Bomb explodes into two parts in the mass ratio 1 : 2 at the point where the vertical component of velocity is zero. If the heavier mass falls vertically down, how far on the ground from the projection point will the lighter mass fall?

1. $4.9\text{m}$
2. $19.6\text{m}$
3. $9.8\text{m}$
4. $2.45\text{m}$

Q. The internal forces can change

1. Linear momentum but not kinetic energy
2. The kinetic energy but not the linear momentum
3. Linear momentum as well as kinetic energy
4. Neither the linear momentum nor the kinetic energy

Q. A bomb at rest explodes into three equal fragments. Two fragments fly off at right angles with velocity $12\text{m/s}$ and $9\text{m/s}$. If the explosion process occurs in $0.1\text{s}$, find out the average force acting on the third particle, if total mass of the bomb is $12\text{kg}$.

1. $600\text{N}$
2. $60\text{N}$
3. $15\text{N}$
4. $1200\text{N}$

Q.

A ball of mass 'm', moving with a velocity 'v' along X-axis, strikes another ball of mass '2m' kept at rest. The first ball comes to rest after collision and the other breaks into two equal pieces. One of the pieces starts moving along Y-axis with a speed $v_1$. What will be the velocity of the other piece?

1. $\sqrt{v^2+v_1^2}$

2. $\sqrt{2v^2+v_1^2}$

3. $\sqrt{v^2-v_1^2}$

4. $\sqrt{\sqrt{v^2+2v_1^2}}$

Q.

A man of mass 'M' having a bag of mass 'm' slips from the roof of a tall building of height 'H' and starts falling vertically (figure). When at a height 'h' from the ground, he notices that the ground below him is pretty hard, but there is a pond at a horizontal distance 'x' from the line of fall. In order to save himself he throws the bag horizontally (with respect to himself) in the direction opposite to the pond. Calculate the minimum horizontal velocity imparted to the bag so that the man lands in the water.

1. $\frac{2mx\sqrt{g}}{M(\sqrt{2H}-\sqrt{2(H-h)})}$

2. $\frac{2mx\sqrt{g}}{M\{\sqrt{2H}-\sqrt{2(H-h)})\}}$

3. $\frac{mx\sqrt{g}}{M\{\sqrt{2H}-\sqrt{2(H-h)})\}}$

4. $\frac{mx}{m\{\sqrt{2gH}-\sqrt{2g(H-h)}\}}$

Q. An explosion breaks a rock into three parts. Two pieces go off at right angles to each other with 1.0 kg piece with a velocity of 12 m/s and other 2.0 kg piece with a velocity of 8 m/s. If the third piece flies off with a velocity of 40 m/s, compute the mass of the third piece.

1. 0.5 kg
2. 3 kg
3. 0.7 kg
4. 0.25kg

Q. If there is no external force, the acceleration of the COM during collision

1. Increases
2. Decreases
3. Remains constant
4. Is zero

Q. A child is standing at one end of a long trolley moving with a speed 'v' on a smooth horizontal track. If the child starts running towards the other end of the trolley with a speed 'u', the centre of mass of the system (trolley + child) will move with a speed

1. zero
2. (v + u)
3. (v - u)
4. v