A ball of mass 10 kg moving with a velocity 10√3 ms^-1 along X-axis, hits another ball of mass 20 kg

Q: Why is momentum conserved even though the second ball disintegrates?

Disintegration is caused by internal forces (like an explosion or structural failure). Internal forces always occur in equal and opposite pairs (Newton's Third Law) and therefore cancel each other out, leaving the total momentum of the isolated system completely unchanged.

Q: Is kinetic energy conserved in this collision?

Not at all! In fact, because the second ball disintegrates, internal potential energy is converted into kinetic energy. If you calculate it, the final kinetic energy ($2500\text{ J}$) is much higher than the initial kinetic energy ($1500\text{ J}$). Momentum conservation is your only reliable tool here.

Q: Why did we assume the second piece moves in the fourth quadrant (negative Y direction)?

The first piece moves along the positive Y-axis. To conserve the initial zero momentum in the Y-direction, the second piece *must* have a velocity component in the negative Y-direction to perfectly cancel it out.

Q: What if the two pieces of the second ball had different masses?

The principle remains exactly the same. You would just substitute the different mass values into the momentum conservation equations: $m_{2a} v_{2ay} + m_{2b} v_{2by} = 0$.

Q: Can we solve this using only the X-axis equation?

Yes, the X-axis equation alone gives $\cos\theta = \frac{\sqrt{3}}{2}$, which implies $\theta = 30^\circ$. However, checking the Y-axis is a great way to verify your answer and ensure the physical setup is mathematically consistent.

Question

A ball of mass $10	ext{ kg}$ moving with a velocity $10\sqrt{3}	ext{ ms}^{-1}$ along X-axis, hits another ball of mass $20	ext{ kg}$, which is at rest. After collision, the first ball comes to rest and the second one disintegrates into two equal pieces. One of the pieces starts moving along Y-axis at a speed of $10	ext{ m/s}$. The second piece starts moving at a speed of $20	ext{ m/s}$ at an angle $	heta$ (degree) with respect to the X-axis. The configuration of pieces after collision is shown in the figure. The value of $	heta$ to the nearest integer is ……… .

Sigma Percentile · Accepted Answer

## The Explosive Collision: Unraveling the Angles of Disintegration

Collisions in physics are rarely just simple bumps. Sometimes, they are explosive events that shatter objects and send pieces flying in multiple directions. In this thrilling problem, we are tasked with tracking the chaotic aftermath of a high-speed impact where a stationary target doesn't just move—it disintegrates!

### Analyzing the Setup: Before the Chaos

Imagine the scene: A solid $10	ext{ kg}$ ball (let's call it Ball A) is hurtling along the X-axis like a cannonball, boasting a velocity of $10\sqrt{3}	ext{ m/s}$. Waiting peacefully ahead is Ball B, a massive $20	ext{ kg}$ sphere completely at rest.

Suddenly, the impact occurs. But instead of a standard bounce, something dramatic happens. Ball A transfers all its forward drive and comes to a dead stop. Ball B, unable to handle the shock, shatters into two perfectly equal $10	ext{ kg}$ pieces.

One piece shoots straight up along the Y-axis at $10	ext{ m/s}$. The other piece dives diagonally downwards at a blistering $20	ext{ m/s}$, making an unknown angle $	heta$ with the X-axis. Our mission? Find that angle.

### The Master Equation: Conservation of Momentum

Even though the collision resulted in an explosion (disintegration), the forces that tore Ball B apart were entirely **internal**. Because there are no external forces acting on our two-ball system, the universe demands that the total linear momentum remains absolutely constant.

Mathematically, the initial momentum vector must equal the final momentum vector:

$$ \vec{p}_i = \vec{p}_f $$

$$ m_1 \vec{u}_1 + m_2 \vec{u}_2 = m_1 \vec{v}_1 + m_{2a} \vec{v}_{2a} + m_{2b} \vec{v}_{2b} $$

Because momentum is a vector, we can slice this complex 2D explosion into two simple 1D problems: one along the X-axis and one along the Y-axis.

### Resolving Along the X-Axis: The Horizontal Chase

Let's look purely at the horizontal motion. Before the crash, only Ball A was moving horizontally. After the crash, Ball A is stopped, the first piece of Ball B is moving purely vertically, and only the second piece of Ball B has a horizontal component.

Equating the initial and final X-momenta:

$$ p_{ix} = p_{fx} $$

$$ 10(10\sqrt{3}) + 20(0) = 10(0) + 10(0) + 10(20 \cos	heta) $$

$$ 100\sqrt{3} = 200 \cos	heta $$

Isolating $\cos	heta$, we get:

$$ \cos	heta = \frac{100\sqrt{3}}{200} = \frac{\sqrt{3}}{2} $$

From our trigonometric knowledge, we instantly recognize that $	heta = 30^\circ$.

### Verifying Along the Y-Axis: The Vertical Balance

In physics, it is always wise to double-check your work. Let's look at the Y-axis. Initially, nothing was moving vertically. The total initial Y-momentum was zero.

Therefore, after the explosion, the upward momentum of the first piece must be perfectly canceled out by the downward momentum of the second piece.

$$ p_{iy} = p_{fy} $$

$$ 0 = 10(10) - 10(20 \sin	heta) $$

$$ 0 = 100 - 200 \sin	heta $$

$$ 200 \sin	heta = 100 $$

$$ \sin	heta = \frac{1}{2} $$

Once again, this confirms that $	heta = 30^\circ$. The physics is perfectly consistent!

### A Surprising Twist: The Energy Audit

We found our angle, but let's dig deeper. Was kinetic energy conserved?

Initial Kinetic Energy:
$$ K_i = \frac{1}{2}(10)(10\sqrt{3})^2 = 1500	ext{ J} $$

Final Kinetic Energy:
$$ K_f = \frac{1}{2}(10)(10)^2 + \frac{1}{2}(10)(20)^2 = 500 + 2000 = 2500	ext{ J} $$

Wait a minute! $2500	ext{ J} > 1500	ext{ J}$. The kinetic energy actually **increased** by $1000	ext{ J}$!

How is this possible? This proves that the disintegration wasn't just a passive breaking apart; it was an **explosive event**. Stored internal potential energy (perhaps chemical or elastic) within Ball B was released during the impact, converting into the extra kinetic energy that sent the pieces flying.

This is why we could only rely on momentum conservation, not energy conservation, to solve the mystery of the angle!

Enter Numerical Value:

Visualized Solution

\text{Visualizing the Collision}

\text{Conservation of Linear Momentum}

\text{Momentum along X-axis}

\text{Solving for } \theta \text{ (X-axis)}

\text{Momentum along Y-axis}

\text{Solving for } \theta \text{ (Y-axis)}

\text{Final Answer}

\text{Energy Analysis (Bonus)}

The Sigma Insight: Conservation of Linear Momentum

Analyzing the Setup

The Master Equation

Resolving Along the X-Axis

Verifying Along the Y-Axis

A Surprising Twist

Similar Questions

A body of mass $1$ kg initially at rest, explodes and breaks into three fragments of masses in the ratio $1 : 1 : 3$ . The two pieces of equal mass fly-off perpendicular to each other with a speed of $30$ m/s each. What is the velocity of the heavier fragment ?