To solve the problem step by step, we will analyze the motion of both particles and calculate the required quantities: acceleration, velocity, and displacement of the center of mass (CM).
### Step 1: Calculate the Acceleration of the Center of Mass (CM)
The acceleration of the center of mass (CM) for two particles can be calculated using the formula:
\[
a_{CM} = \frac{m_1 a_1 + m_2 a_2}{m_1 + m_2}
\]
Where:
- \(m_1 = 1 \, \text{kg}\) (mass of the first particle)
- \(m_2 = 2 \, \text{kg}\) (mass of the second particle)
- \(a_1 = -g = -10 \, \text{m/s}^2\) (acceleration of the first particle, upwards)
- \(a_2 = g = 10 \, \text{m/s}^2\) (acceleration of the second particle, downwards)
Substituting the values:
\[
a_{CM} = \frac{1 \cdot (-10) + 2 \cdot 10}{1 + 2} = \frac{-10 + 20}{3} = \frac{10}{3} \, \text{m/s}^2
\]
Since both accelerations are directed downwards, we take \(g\) as positive for the downward direction. Hence, the acceleration of the center of mass is:
\[
a_{CM} = 10 \, \text{m/s}^2
\]
### Step 2: Calculate the Velocity of the Center of Mass (CM)
To find the velocity of the CM after 2 seconds, we first calculate the velocities of both particles.
**For Particle 1 (1 kg):**
Using the equation of motion:
\[
v_1 = u_1 + a_1 t
\]
Where:
- \(u_1 = 60 \, \text{m/s}\) (initial velocity)
- \(a_1 = -10 \, \text{m/s}^2\) (acceleration)
- \(t = 2 \, \text{s}\)
Substituting the values:
\[
v_1 = 60 + (-10) \cdot 2 = 60 - 20 = 40 \, \text{m/s}
\]
**For Particle 2 (2 kg):**
Using the equation of motion:
\[
v_2 = u_2 + a_2 t
\]
Where:
- \(u_2 = 0 \, \text{m/s}\) (initial velocity)
- \(a_2 = 10 \, \text{m/s}^2\) (acceleration)
- \(t = 2 \, \text{s}\)
Substituting the values:
\[
v_2 = 0 + 10 \cdot 2 = 20 \, \text{m/s}
\]
Now, we calculate the velocity of the center of mass:
\[
v_{CM} = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}
\]
Substituting the values:
\[
v_{CM} = \frac{1 \cdot 40 + 2 \cdot (-20)}{1 + 2} = \frac{40 - 40}{3} = 0 \, \text{m/s}
\]
### Step 3: Calculate the Displacement of the Center of Mass (CM)
**For Particle 1 (1 kg):**
Using the equation of motion:
\[
s_1 = u_1 t + \frac{1}{2} a_1 t^2
\]
Substituting the values:
\[
s_1 = 60 \cdot 2 + \frac{1}{2} \cdot (-10) \cdot (2)^2 = 120 - 20 = 100 \, \text{m}
\]
**For Particle 2 (2 kg):**
Using the equation of motion:
\[
s_2 = u_2 t + \frac{1}{2} a_2 t^2
\]
Substituting the values:
\[
s_2 = 0 \cdot 2 + \frac{1}{2} \cdot 10 \cdot (2)^2 = 0 + 20 = 20 \, \text{m}
\]
Now, we calculate the displacement of the center of mass:
\[
s_{CM} = \frac{m_1 s_1 + m_2 s_2}{m_1 + m_2}
\]
Substituting the values:
\[
s_{CM} = \frac{1 \cdot 100 + 2 \cdot (-20)}{1 + 2} = \frac{100 - 40}{3} = \frac{60}{3} = 20 \, \text{m}
\]
### Final Results
- Acceleration of CM: \(10 \, \text{m/s}^2\)
- Velocity of CM: \(0 \, \text{m/s}\)
- Displacement of CM: \(20 \, \text{m}\)
### Matching the Results
- Acceleration of CM (A) matches with (Q) 10 SI unit.
- Velocity of CM (B) matches with (P) 0.
- Displacement of CM (C) matches with (R) 20 SI unit.
