A 6 kg box is travelling on ice at a speed of 9 m/s when a 12 kg packet is gently placed on it. The velocity will now be
From Principal of conservation of momentum
m 1 v 1 = (m 1 + m 2 ) v
6 Ć 9 = ( 6 + 12) Ć v ā v = 3 m/s
A ball, moving with a speed v towards north, collides with an identical ball, moving with a speed v towards east. After collision the two balls stick together and move towards north-east. The speed of the combination is
Taking both the balls as one system
mv i + mv j= 2mĆ v
so, v= v/2 i + v/2Ā j
so, |v|= v/ā2
A bomb of mass M at rest explodes into three pieces, two of which of mass M/4 each, are thrown off in perpendicular directions with speeds of 3 m/s and 4 m/s. The third piece is thrown off with a speed
As the bomb was initially at rest and no external force acts on it total momentum of the bomb should remain constant.
so, (m/4) 3 i +(m/4) 4 j + (m/2) v1 = 0
v1 = 3/2 i + 4/2 j
|V1|= 2.5 m/s
Assertion: If kinetic energy of a system of particles is zero, then linear momentum of system must be zero.
Reason: If linear momentum of a system of particles is zero, then kinetic energy of system must be zero.
If the kinetic energy of a system is zero, the speed of each particle must be zero, meaning the total linear momentum is also zero. If the total linear momentum is zero, particles can still be moving in opposite directions, resulting in a non-zero kinetic energy. Thus, the Assertion is true but the Reason is false.
A body of mass 2 kg moving with velocity \(5 \text{ m s}^{-1}\) collides with a body at rest of mass 3 kg and sticks to it. Now the combined mass starts moving. The final velocity of whole mass is
Using the law of conservation of linear momentum: \(m_1 u_1 + m_2 u_2 = (m_1 + m_2) v_f\). Substituting the values: \(2(5) + 3(0) = (2 + 3) v_f ā 10 = 5 v_f ā v_f = 2 \text{ m s}^{-1}\).
Assertion (A): A body with negative energy cannot have linear momentum.
Reason (R): Magnitude of linear momentum can be negative.
Total mechanical energy \(E = K + U\) can be negative if potential energy \(U\) is negative and larger in magnitude than kinetic energy \(K\).
However, kinetic energy \(K = \frac{1}{2}mv^2\) is always non-negative, implying momentum exists. The magnitude of linear momentum \(|\vec{p}| = mv\) is always non-negative.
Therefore, both Assertion and Reason are false.
Assertion (A): Two objects are moving towards each other due to mutual attraction. The kinetic energy of the system remains constant.
Reason (R): Total linear momentum of the system consisting both the objects remain constant even in the presence of external forces.
Assertion (A) is false; as objects move towards each other due to mutual attraction, their potential energy decreases, and kinetic energy increases, so it does not remain constant.
Reason (R) is false; total linear momentum of a system is conserved only in the absence of external forces, not 'even in the presence of external forces'. Therefore, both (A) and (R) are false.