Assertion (A): Maximum energy loss occurs when the particles get stuck together as a result of collision.
Reason (R): A point particle of mass (m\) moving with speed (v\) collides with stationary point particle of mass (M\). Then the maximum energy loss possible is given \( \frac{m}{(m+M)}\left(\frac{1}{2}mv^2\right)\).
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Assertion (A): Maximum kinetic energy loss occurs in a perfectly inelastic collision where particles stick together. So, (A) is true.
Reason (R): For a perfectly inelastic collision between mass (m\) (velocity (v\)) and stationary mass (M\), the energy loss is ( \Delta K = \frac{M}{(m+M)}\left(\frac{1}{2}mv^2\right)\). The given formula in (R) is incorrect.
So, (R) is false. Therefore, (A) is true and (R) is false. Option (3) is correct.
Assertion (A): In case of bullet fired from a gun, the ratio of kinetic energy of gun and bullet is equal to ratio of masses of bullet and gun.
Reason (R): In firing of bullet, linear momentum of system is conserved.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Reason (R): For the bullet-gun system, the forces causing the bullet to fire are internal. Thus, linear momentum of the system is conserved. So, (R) is true.
Assertion (A): Let (m\) and (M\) be masses of bullet and gun, (v\) and (V\) their velocities. By momentum conservation, (mv = MV\). The ratio of kinetic energies is \( \frac{K_g}{K_b} = \frac{\frac{1}{2}MV^2}{\frac{1}{2}mv^2} = \frac{M(mv/M)^2}{mv^2} = \frac{m}{M}\). So, (A) is true.
(R) correctly explains (A) as the kinetic energy ratio is derived directly from momentum conservation. Option (1) is correct.
Assertion (A): The centre of mass of a system of two particles is closer to the heavier particle.
Reason (R): Algebraic sum of mass moments about centre of mass is zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
View Answer
For a two-particle system, the center of mass \( R_{CM} \) is defined such that the sum of mass moments about it is zero: \( m_1r_1 = m_2r_2 \). If \( m_1 > m_2 \), then \( r_1 < r_2 \), meaning the COM is closer to the heavier particle.
Thus both A and R are true, and R explains A.
The centre of mass of a system of particles depends on
1. Position of the particles
2. Relative distance between the particles
3. Masses of the particles
4. All of these
View Answer
The position of the centre of mass of a system of particles is defined as \( \vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \). It clearly depends on individual masses, their coordinates (positions), and consequently the relative distances between them.
Consider the given statements and choose the correct option that follows:
Statement 1: During a collision the total linear momentum of system is conserved at each instant of collision.
Statement 2: During a collision the kinetic energy conservation holds always.
Based on above information, pick the correct option.
1. Both statements (1) and (2) are true
2. Both statements (1) and (2) are false
3. Statement (1) is true but (2) is false
4. Statement (1) is false but (2) is true
View Answer
Total linear momentum is conserved at each instant of collision because no external forces act. Kinetic energy, however, is not conserved during the period of deformation, and is conserved after only in perfectly elastic collisions. Thus, Statement 1 is true and Statement 2 is false.