A projectile is moving at `20 m s^(-1)` at its highest point where it breaks into equal parts due to an internal explosion. Just after explosion, one part moves vertically up at `30 ms^(-1)` with respect to the ground. Then the other part will move at :

Options

A. `20 m//s`
B. `10 sqrt(13) m//s`
C. `50 m//s`
D. `30 m//s`

Explanation

To solve the problem step by step, we will analyze the situation using the principles of conservation of momentum.

### Step 1: Understand the Initial Conditions
At the highest point of its trajectory, the projectile is moving horizontally with a velocity of \(20 \, \text{m/s}\). This means that its initial momentum can be expressed as:
\[ \text{Initial Momentum} = m \cdot \vec{V} = m \cdot (20 \hat{i}) \, \text{kg m/s} \]
where \(m\) is the mass of the projectile.

### Step 2: Analyze the Explosion
When the projectile explodes, it breaks into two equal parts, each with mass \(\frac{m}{2}\). One part moves vertically upward with a velocity of \(30 \, \text{m/s}\) (in the \(\hat{j}\) direction). We can express this velocity as:
\[ \vec{V_1} = 30 \hat{j} \, \text{m/s} \]

### Step 3: Set Up the Conservation of Momentum Equation
Since there are no external forces acting on the system, the total momentum before the explosion must equal the total momentum after the explosion. Therefore, we can write:
\[ \text{Initial Momentum} = \text{Final Momentum} \]
This gives us:
\[ m \cdot (20 \hat{i}) = \frac{m}{2} \cdot (30 \hat{j}) + \frac{m}{2} \cdot \vec{V_2} \]

### Step 4: Simplify the Equation
We can cancel \(m\) from both sides (assuming \(m \neq 0\)):
\[ 20 \hat{i} = \frac{1}{2} (30 \hat{j}) + \frac{1}{2} \vec{V_2} \]
Multiplying through by 2 to eliminate the fractions:
\[ 40 \hat{i} = 30 \hat{j} + \vec{V_2} \]

### Step 5: Solve for \(\vec{V_2}\)
Rearranging the equation to isolate \(\vec{V_2}\):
\[ \vec{V_2} = 40 \hat{i} - 30 \hat{j} \]

### Step 6: Find the Magnitude of \(\vec{V_2}\)
To find the magnitude of the velocity of the second part, we calculate:
\[ |\vec{V_2}| = \sqrt{(40)^2 + (-30)^2} = \sqrt{1600 + 900} = \sqrt{2500} = 50 \, \text{m/s} \]

### Final Answer
The other part will move at a speed of \(50 \, \text{m/s}\).

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