© All Rights reserved @ LearnWithDash
Step-by-Step Solution
Step 1: Understand the Given Data
Two particles have masses 4 g and 16 g, respectively. They are moving with the same kinetic energy (KE). The ratio of their linear momenta is given as n : 2. We are asked to find the value of n.
Step 2: Write the Relation Between Momentum and Kinetic Energy
For a particle of mass m and kinetic energy KE , the linear momentum p is given by:
p = \sqrt{2 m \cdot KE}.
Since both particles have the same kinetic energy, we can compare their momenta directly.
Step 3: Express the Momentum Ratio
Let the momentum of the 4 g particle be p_1 and that of the 16 g particle be p_2 . Then:
\frac{p_1}{p_2} = \sqrt{\frac{m_1}{m_2}},
because KE is the same for both.
Substitute m_1 = 4\,\text{g} and m_2 = 16\,\text{g} :
\frac{p_1}{p_2} = \sqrt{\frac{4}{16}} = \sqrt{\frac{1}{4}} = \frac{1}{2}.
Step 4: Relate the Ratio to n : 2
The problem states that p_1 : p_2 = n : 2 , which means
\frac{p_1}{p_2} = \frac{n}{2}.
We found above that \frac{p_1}{p_2} = \frac{1}{2}. Hence,
\frac{n}{2} = \frac{1}{2}.
Step 5: Solve for n
From the equation above, we immediately get
n = 1.
Thus, the required ratio of the magnitudes of linear momentum is 1 : 2.
