Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason

Q: Why does the small mass get exactly twice the speed? That feels like it's violating conservation of energy!

It doesn't violate energy conservation at all! While the small mass gets a high velocity ($2u$), its mass ($m$) is so tiny that its total kinetic energy ($\frac{1}{2}m(2u)^2$) is still just a tiny fraction of the massive body's total energy. The massive body loses an imperceptibly small amount of speed, which perfectly accounts for the energy transferred to the small mass.

Q: Does the large mass $M$ lose any energy at all?

Technically, yes. It must lose a tiny bit of energy to accelerate the small mass $m$. Its final velocity $v_1$ is actually slightly less than $u$. However, because $M$ is astronomically larger than $m$, this change in velocity is so infinitesimally small that for all practical calculation purposes, we approximate $v_1 \approx u$.

Q: What would happen if the small mass $m$ was moving towards the large mass $M$ with speed $u$ instead?

If the small mass hits the stationary large mass, the small mass will bounce back with almost exactly its initial speed ($-u$), and the large mass will barely move. It's like throwing a bouncy ball at a brick wall!

Q: Why do we need to use both momentum and energy conservation? Can't we just use one?

You need both! We have two unknown variables after the collision: the final velocity of the large mass ($v_1$) and the final velocity of the small mass ($v_2$). In algebra, to solve for two unknowns, you need two independent equations. Momentum gives us one, and kinetic energy gives us the other.

Q: Is this a perfectly elastic collision in real life?

In the macroscopic real world, perfectly elastic collisions are an idealization. Some energy is always lost to sound, heat, or permanent deformation (like a car crash). However, collisions between subatomic particles or highly rigid objects (like billiard balls or steel spheres) can be very close to perfectly elastic.

Question

Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R.
**Assertion (A)** Body $P$ having mass $M$ moving with speed $u$ has head-on collision elastically with another body $Q$ having mass $m$ initially at rest. If $m \ll M$, body $Q$ will have a maximum speed equal to $2u$ after collision.
**Reason (R)** During elastic collision, the momentum and kinetic energy are both conserved.
In the light of the above statements, choose the most appropriate answer from the options given below.

Options:

(A) A is not correct but R is correct.

(B) Both A and R are correct but R is not the correct explanation of A.

(C) Both A and R are correct and R is the correct explanation of A.

(D) A is correct but R is not correct.

Sigma Percentile · Accepted Answer

### The Setup: David vs Goliath

Imagine a scenario straight out of a physics thought experiment: a massive, unstoppable object (let's call it Goliath, or body $P$ with mass $M$) is barreling down a frictionless path with a speed $u$. Sitting completely still in its path is a tiny, almost massless object (David, or body $Q$ with mass $m$). The problem states that $M \gg m$, meaning the mass of $P$ is astronomically larger than the mass of $Q$.

They are about to undergo a perfectly elastic head-on collision. Our goal is to determine what happens to the tiny body $Q$ after this cosmic impact. Will it be crushed? Will it move? If so, how fast? The Assertion claims it will shoot off with a speed of $2u$. Let's dive into the fundamental laws of the universe to see if this holds true.

### The Laws of the Universe: Momentum and Energy

In the realm of classical mechanics, a perfectly elastic collision is governed by two unbreakable rules. First, because there are no external forces acting on our two-body system, the **Law of Conservation of Linear Momentum** dictates that the total momentum before the crash must equal the total momentum after the crash.

Mathematically, we write this as:

$$Mu + m(0) = Mv_1 + mv_2$$

Here, $v_1$ and $v_2$ are the final velocities of the massive body $P$ and the tiny body $Q$, respectively. We can rearrange this equation to group the terms associated with the massive body $M$:

$$M(u - v_1) = mv_2$$

$$\frac{M(u - v_1)}{m} = v_2 \quad \dots (1)$$

Second, the "elastic" nature of the collision means that no kinetic energy is lost to heat, sound, or deformation. The **Law of Conservation of Kinetic Energy** applies perfectly:

$$\frac{1}{2}Mu^2 + 0 = \frac{1}{2}Mv_1^2 + \frac{1}{2}mv_2^2$$

### The Mathematical Symphony

Let's simplify our energy equation. We can multiply the entire equation by 2 to eliminate the fractions, and again, group the terms with $M$ on one side:

$$M(u^2 - v_1^2) = mv_2^2$$

Using the algebraic identity for the difference of squares, we can expand the left side:

$$M(u - v_1)(u + v_1) = mv_2^2 \quad \dots (2)$$

Now, we have a beautiful opportunity for substitution. Notice that the term $M(u - v_1)$ appears in both our momentum equation (1) and our expanded energy equation (2). Let's substitute $mv_2$ from equation (1) into equation (2):

$$mv_2(u + v_1) = mv_2^2$$

Assuming the tiny body $Q$ actually moves after the collision (so $v_2 
eq 0$), we can divide both sides by $mv_2$. This leaves us with an incredibly elegant and simple kinematic relationship:

$$u + v_1 = v_2$$

### The Grand Conclusion: The Slingshot Effect

This is where the physical reality of our $M \gg m$ approximation comes into play. Think back to our analogy: a massive truck hitting a stationary ping-pong ball. Does the truck slow down noticeably after hitting the ball? Of course not. Its inertia is simply too great.

Mathematically, because $M$ is so much larger than $m$, the final velocity of the massive body ($v_1$) is practically identical to its initial velocity ($u$).

$$	herefore v_1 \approx u$$

Now, let's substitute this physical insight back into our simplified kinematic equation:

$$u + u = v_2$$

$$v_2 = 2u$$

The math confirms our intuition! The tiny body $Q$ is launched forward at exactly **twice the initial speed** of the massive body $P$.

Looking back at the original question, the Assertion (A) correctly states that the maximum speed of $Q$ is $2u$. The Reason (R) correctly states that both momentum and kinetic energy are conserved in an elastic collision. Furthermore, it was precisely the application of these two conservation laws that allowed us to prove the Assertion. Therefore, both statements are correct, and the Reason is the correct explanation for the Assertion.

Select Answer:

Visualized Solution

$Visualizing the Collision$

$Conservation of Momentum$

$Momentum Equation$

$Conservation of Kinetic Energy$

$Energy Equation$

$Solving for Velocities$

$Applying the Approximation$

$Final Speed of Body Q$

$Conclusion$

The Sigma Insight: Head-on Collision

The Setup

The Laws of the Universe

The Mathematical Symphony

The Grand Conclusion

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Visualizing the Collision

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conservation of Momentum

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Momentum Equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conservation of Kinetic Energy

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Energy Equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Solving for Velocities

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Applying the Approximation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Speed of Body Q

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Conclusion