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### Problem Summary:
A watermelon is blown into three pieces. Two pieces have equal mass \( m \) and move perpendicularly to each other with a velocity \( V = 36 \, \text{m/s} \) (one along the x-axis and the other along the y-axis). The third piece, which is three times more massive \( 3m \), flies off at an angle \( \theta \) with speed \( V_f \).

We need to solve the following:

### (a) **Expression for the speed \( V_f \) of the larger piece:**

Using the conservation of momentum, the total momentum before and after the explosion must be zero (as the watermelon was initially at rest). 

- The x-component of momentum:
  \[
  mV - 3mV_f \cos(\theta) = 0 \quad \Rightarrow \quad V_f \cos(\theta) = \frac{V}{3}
  \]

- The y-component of momentum:
  \[
  mV - 3mV_f \sin(\theta) = 0 \quad \Rightarrow \quad V_f \sin(\theta) = \frac{V}{3}
  \]

Now, combining these two components using Pythagoras’ theorem:
\[
V_f = \sqrt{\left( \frac{V}{3} \right)^2 + \left( \frac{V}{3} \right)^2} = \frac{V}{3} \sqrt{2}
\]

Thus, the speed of the larger piece is:
\[
V_f = \frac{V \sqrt{2}}{3}
\]

### (b) **Numeric value for \( V_f \):**

Substituting \( V = 36 \, \text{m/s} \) into the equation:
\[
V_f = \frac{36 \times \sqrt{2}}{3} \approx \frac{36 \times 1.414}{3} = 16.97 \, \text{m/s}
\]
Thus, the speed of the larger piece is approximately **16.97 m/s**.

### (c) **Angle \( \theta \) of the larger piece:**

The angle can be found from the ratio of the momentum components:
\[
\tan(\theta) = \frac{V_y}{V_x} = \frac{mV}{mV} = 1
\]
Therefore:
\[
\theta = \tan^{-1}(1) = 45^\circ
\]

Thus, the angle at which the larger piece travels is **45 degrees**.

---

Would you like further details or explanations on any of these steps? 

Here are 5 related questions to expand your understanding:
1. How does the conservation of momentum apply to explosions in different directions?
2. What role does mass play in determining the velocity of fragments after an explosion?
3. How would the solution change if the initial velocity of the watermelon were not zero?
4. What would happen to the final velocities if the two smaller pieces had different masses?
5. How can we generalize the use of trigonometric functions in 2D momentum problems?

**Tip:** Always check if the system is initially at rest when applying conservation of momentum to explosions or collisions.

A watermelon is blown into three pieces by a firecracker. Two pieces of equal mass m fly away perpendicularly, one along the x-axis and the other along the y-axis, each with a speed V = 36 m/s. The third piece has three times the mass of the other pieces. Write an expression for the speed of the larger piece in terms of V, calculate the numeric value for its speed, and determine the angle at which it travels with respect to the -y axis.

Learn how to apply conservation of momentum to a problem where a watermelon explodes into three pieces. This physics problem involves calculating the speed and angle of a larger fragment using conservation principles and trigonometry. Includes detailed solution with key formulas and theorems.

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