Animated Solution for Physics - Work, Energy, and Power: Two particles having masses $4\text{ g}$ and $16\text{ g}$ respectively are moving with equal kinetic energies. The ratio of the magnitudes of their linear momentum is $n:2$. The value of $n$ will be ............ .

Imagine you are observing two particles, let's call them $A$ and $B$, zooming through space. Particle $A$ is a lightweight contender, weighing in at just $4\text{ g}$. On the other hand, particle $B$ is the heavyweight champion of this scenario, with a mass of $16\text{ g}$.

Despite their massive weight difference, the universe has granted them a beautiful symmetry: they both possess the exact same kinetic energy.

This is where our physics intuition must kick in. If a lighter particle has the same kinetic energy as a heavier one, it must be moving significantly faster to compensate for its lack of mass. But the question isn't asking about their velocities; it's asking about their linear momenta.

To solve this, we need a mathematical bridge that connects kinetic energy ($K$) directly to linear momentum ($p$).

You might immediately think of $K=\frac{1}{2}m{v}^2$ and $p=mv$. While you can certainly use these two equations to find the answer, there is a much more elegant and direct relationship. By substituting $v=\frac{p}{m}$ into the kinetic energy formula, we get the golden equation:

This single equation is the key to unlocking the entire problem. It beautifully ties together the three quantities we care about: kinetic energy, momentum, and mass.

The problem explicitly states that the kinetic energies of both particles are equal. Let's write this down mathematically:

Now, we substitute our golden equation into this equality for both particles:

This is our raw setup. We have successfully translated the physical situation into a pure mathematical equation.

Now comes the fun part: plugging in the numbers. We know $m_A=4\text{ g}$ and $m_B=16\text{ g}$.

A quick pro-tip: You might be tempted to convert these masses into kilograms to stick to SI units. However, because we are dealing with an equation where mass appears in the denominator on both sides, any conversion factor would simply cancel out! So, save yourself the time and effort, and just plug in the values in grams.

Simplifying the denominators, we get:

Let's rearrange this to find the ratio of their momenta squared. We want to isolate $\frac{p_A^2}{p_B^2}$:

Simplifying the fraction on the right side:

We are almost there! We have the ratio of their squared momenta, but we need the ratio of the magnitudes of the momenta themselves. To get this, we simply take the square root of both sides:

So, the ratio of the magnitude of their linear momenta is $1:2$.

The problem states that this ratio is $n:2$. By directly comparing our result with the given format, the answer reveals itself:

And there you have it! By leveraging the direct relationship between kinetic energy and momentum, we bypassed the need to calculate velocities and arrived at the solution with elegant simplicity.