A ball of mass 10 kg moving with a velocity 10√3 m/s along the X-axis, hits another ball of mass 20

Q: Why is momentum conserved in this collision?

Momentum is conserved because there are no net external forces acting on the system of the two balls. The forces during the collision and explosion are purely internal.

Q: Why don't we use conservation of kinetic energy?

The problem states the second ball 'disintegrates' or explodes. Explosions involve the release of internal energy (like chemical or elastic potential energy) into kinetic energy, making the collision inelastic. Kinetic energy is not conserved.

Q: What happens to the momentum in the Y-direction?

Initially, there is no momentum in the Y-direction. After the collision, the upward momentum of one piece perfectly cancels the downward momentum of the other piece, keeping the total Y-momentum zero.

Q: How do we know the masses of the pieces?

The problem explicitly states that the 20 kg ball disintegrates into 'two equal pieces.' Therefore, each piece has a mass of 10 kg.

Q: Why is the X-velocity of the first piece zero?

The problem states that the first piece moves 'along the Y-axis.' This means its entire velocity is vertical, and its horizontal (X-axis) component is exactly zero.

Question

A ball of mass 10 kg moving with a velocity $10\sqrt{3}$ m/s along the X-axis, hits another ball of mass 20 kg which is at rest. After the collision, first ball comes to rest while the second ball disintegrates into two equal pieces. One piece starts moving along Y-axis with a speed of 10 m/s. The second piece starts moving at an angle of 30° with respect to the X-axis. The velocity of the ball moving at 30° with X-axis is $x$ m/s. The configuration of pieces after collision is shown in the figure below. The value of $x$ to the nearest integer is .......... .

Sigma Percentile · Accepted Answer

### Cosmic Billiards: Unraveling an Explosive Collision

Imagine a game of cosmic billiards! A $10 	ext{ kg}$ ball comes rushing in at a blistering $10\sqrt{3} 	ext{ m/s}$ and smashes into a stationary $20 	ext{ kg}$ ball. Boom! The first ball stops dead in its tracks, and the second ball shatters into two equal $10 	ext{ kg}$ pieces flying off in completely different directions.

This isn't just a chaotic explosion; it's a beautifully choreographed dance governed by one of the most fundamental laws of the universe: **The Conservation of Linear Momentum**.

### The Master Equation

Since there are no external forces acting on our system during this explosive collision, the total momentum must be conserved. This means the momentum before the crash exactly equals the momentum after, in every single direction!

Let's focus on the horizontal X-axis. Initially, only the first ball is moving. After the collision, the first ball stops, one piece flies straight up along the Y-axis (meaning it has zero horizontal velocity), and the other piece moves at a $30^\circ$ angle.

Let's write down the momentum equation for the X-direction:

$$ \sum P_{ix} = \sum P_{fx} $$

$$ m_1 u_{1x} + m_2 u_{2x} = m_1 v_{1x} + m_{2a} v_{2ax} + m_{2b} v_{2bx} $$

### Final Calculation

Plugging in the masses and velocities, the initial momentum is $10 	imes 10\sqrt{3}$. The final horizontal momentum comes entirely from the second piece, which is $10 	imes v_3 \cos 30^\circ$.

$$ 10(10\sqrt{3}) + 20(0) = 10(0) + 10(0) + 10(v_3 \cos 30^\circ) $$

$$ 100\sqrt{3} = 10 v_3 \left(\frac{\sqrt{3}}{2}ight) $$

Now, we just solve for $v_3$. The $\sqrt{3}$ cancels out on both sides, and we are left with a simple equation:

$$ 100 = 5 v_3 \implies v_3 = 20 	ext{ m/s} $$

So, the velocity of the piece moving at $30^\circ$ is $20 	ext{ m/s}$. The question asks for this value as $x$, which means $x$ is exactly **20**. A perfect integer!

### The Sanity Check

As a quick sanity check, you can verify the momentum in the Y-direction. The upward piece has a momentum of $10 	imes 10 = 100$. The downward piece has a vertical momentum of $10 	imes 20 \sin 30^\circ = 100$! They perfectly cancel each other out, matching the initial zero vertical momentum. Physics works beautifully!

Enter Numerical Value:

Visualized Solution

Visual Anchor

Logic Bridge

Raw Setup (X-Direction)

Atomic Compute (Substitution)

Atomic Compute (Execution)

Final Answer

The Way Forward

The Sigma Insight: Conservation of Linear Momentum

Cosmic Billiards

The Master Equation

Final Calculation

The Sanity Check

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