Understanding Kinetic Energy and Speed

This question asks us to compare the speeds of two steel balls with different masses but the same kinetic energy. To solve this, we need to understand the concept of kinetic energy and its relationship with mass and speed.

What is Kinetic Energy?

Kinetic energy is the energy possessed by an object due to its motion. The formula for kinetic energy (KE) is given by:

$\text{KE} = \frac{1}{2}mv^2$

Where:

• $m$ is the mass of the object
• $v$ is the speed (or velocity) of the object

From the formula, we can see that kinetic energy depends on both the mass and the square of the speed of the object.

Comparing Speeds with Equal Kinetic Energy

We are given two steel balls with different masses: one is 5 kg and the other is 10 kg. Both balls possess equal kinetic energy. Let's denote the mass and speed of the first ball (5 kg) as $m_1$ and $v_1$, and the mass and speed of the second ball (10 kg) as $m_2$ and $v_2$.

• Mass of ball 1, $m_1 = 5$ kg
• Mass of ball 2, $m_2 = 10$ kg
• Kinetic energy of ball 1 = Kinetic energy of ball 2 (Let's call this $KE_{equal}$)

Using the kinetic energy formula for both balls:

$\text{KE}_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} (5) v_1^2$

$\text{KE}_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} (10) v_2^2$

Since $\text{KE}_1 = \text{KE}_2$, we can set the expressions equal to each other:

$\frac{1}{2} (5) v_1^2 = \frac{1}{2} (10) v_2^2$

We can cancel the $\frac{1}{2}$ from both sides:

$5 v_1^2 = 10 v_2^2$

Now, let's find the relationship between $v_1$ and $v_2$. Divide both sides by 5:

$v_1^2 = 2 v_2^2$

To find the relationship between the speeds, take the square root of both sides:

$v_1 = \sqrt{2 v_2^2} = \sqrt{2} \cdot \sqrt{v_2^2} = \sqrt{2} v_2$

Since $\sqrt{2}$ is approximately 1.414, the equation is approximately:

$v_1 \approx 1.414 v_2$

This tells us that the speed of the 5 kg ball ($v_1$) is $\sqrt{2}$ times the speed of the 10 kg ball ($v_2$). Since $\sqrt{2} > 1$, this means $v_1 > v_2$. The ball with the smaller mass is moving faster when both have the same kinetic energy.

Analyzing the Options

Let's look at the given options based on our findings:

• The 10 kg ball is moving faster: This contradicts our result that the 5 kg ball is faster.
• Both the balls are moving at the same speed: This is incorrect as we found $v_1 = \sqrt{2} v_2$, and $\sqrt{2} \neq 1$.
• Kinetic energy does not depend upon the speed of the body: This is incorrect. The formula $\text{KE} = \frac{1}{2}mv^2$ clearly shows that KE depends on speed.
• The 5 kg ball is moving faster: This matches our conclusion that $v_1 > v_2$.

Conclusion on Ball Speed

Based on the kinetic energy formula and the fact that both balls have equal kinetic energy, the ball with the smaller mass (5 kg) must be moving faster than the ball with the larger mass (10 kg) to compensate for the mass difference and have the same kinetic energy.

Revision Table: Kinetic Energy Concepts

Additional Information: Energy and Motion

Kinetic energy is a scalar quantity, meaning it only has magnitude, not direction. It's one form of mechanical energy. Another important form is potential energy, which is stored energy, for example, due to an object's position (gravitational potential energy) or shape (elastic potential energy).

The relationship between kinetic energy, mass, and speed highlights an inverse square relationship between mass and speed when kinetic energy is held constant. This means a small change in speed has a larger impact on kinetic energy than a proportional change in mass because speed is squared in the formula.

In physics problems involving motion, energy concepts are often used alongside concepts like force, work, and momentum. Understanding how kinetic energy changes with mass and velocity is fundamental to solving many dynamics problems.