Problem 18 - Master Resource Book in JEE Main Physics

Question

The maximum vertical distance through which a full dressed astronaut can jump on the earth is \(0.5 \mathrm{~m}\). Estimate the maximum vertical distance through which he can jump on the moon, which has a mean density \(2 / 3 \mathrm{rd}\) that of earth and radius one quarter that of the earth (a) \(1.5 \mathrm{~m}\) (b) \(3 \mathrm{~m}\) (c) \(6 \mathrm{~m}\) (d) \(7.5 \mathrm{~m}\)

Step-by-Step Solution

Answer

The maximum jump height on the moon is 7.5 m, so the answer is (d).

Step 1: Understand the Problem

We need to find out how high an astronaut can jump on the moon compared to Earth. Given that the maximum jump height on Earth is 0.5 m, we need to consider the difference in gravitational forces between Earth and the Moon.

Step 2: Determine Gravitational Force on Earth

The gravitational force on Earth can be calculated using the formula for gravitational force: \[g = \frac{G M}{R^2}\]where \(G\) is the gravitational constant, \(M\) is the Earth's mass, and \(R\) is the Earth's radius.

Step 3: Calculate Gravitational Force on Moon

The Moon's radius is one quarter that of Earth, \[R_{\text{moon}} = \frac{1}{4} R_{\text{earth}}\]The Moon's mean density is two-thirds that of Earth: \[\rho_{\text{moon}} = \frac{2}{3} \rho_{\text{earth}}\]The Moon's gravitational force becomes: \[g_{\text{moon}} = \frac{G \cdot M_{\text{moon}}}{(\frac{1}{4}R)^2}\]Given \(M_{\text{moon}} = \frac{2}{3} \times M_{\text{earth}}\), substitute into the equation.

Step 4: Simplify Gravity Expression

Let's express the Moon's gravity in terms of Earth's gravity:\[g_{\text{moon}} = \frac{G \cdot \frac{2}{3}M_{\text{earth}}}{\left(\frac{R_{\text{earth}}}{4}\right)^2} = \frac{G \cdot \frac{2}{3}M}{\frac{R^2}{16}}\]\[g_{\text{moon}} = \frac{32}{3} \frac{G M}{R^2} = \frac{32}{3} g\]Plugging in \(g\): \[g_{\text{moon}} = \frac{8}{9} g_{\text{earth}}\]

Step 5: Calculate the Jump Height on the Moon

The jump height on the Moon can be expressed as: \[H_{\text{moon}} = \frac{g_{\text{earth}}}{g_{\text{moon}}} \cdot H_{\text{earth}}\]Substitute the values from the previous step: \[H_{\text{moon}} = \left(\frac{g_{\text{earth}}}{\frac{8}{9}g_{\text{earth}}}\right) \times 0.5 \]\[H_{\text{moon}} = \left(\frac{9}{8}\right) \times 0.5 = 7.5 \]

Step 6: Confirm Solution

After solving, the maximum vertical jump height on the Moon comes out to be 7.5 m. Verify with the given options to select the correct one.

Key Concepts

Moon gravity

As you might have heard, the Moon's gravity is much weaker than Earth's gravity. This is primarily due to its smaller mass and size. Gravity is the force that pulls objects toward the center of a massive body, like a planet or moon. On the Moon, this force is significantly less because:

• The Moon has a lower mass.
• It is considerably smaller, with only about a quarter of Earth's radius.

These factors result in gravity that is roughly one-sixth of Earth's. Having less gravitational pull on the Moon allows objects and people to move more easily and jump higher compared to their potential on Earth. With Moon gravity, you can expect the same force exerted on the Earth to have greater effects in terms of movement height or force exerted.

Jump height calculation

When calculating how high an astronaut can jump on the Moon, we need to start with how high they jump on Earth, which we know is 0.5 meters. To figure out their jump height on the Moon, we need to consider the gravitational difference.Given the lower gravity on the Moon, the calculation uses the relationship between the gravitational forces. The jump height on the Moon can be determined using the formula: \[H_{\text{moon}} = \frac{g_{\text{earth}}}{g_{\text{moon}}} \cdot H_{\text{earth}}\]Since the Moon's gravity is \(\frac{8}{9}\) of Earth's gravity, the equation simplifies. This calculation tells us that heights are inversely proportional to the gravitational forces. So:

• On Earth: 0.5 meters
• On the Moon: 7.5 meters

This consideration accounts for the change in gravitational pull and results in a much higher jump capability on the Moon than on Earth.

Density ratio

The density of an object is its mass per unit volume. Understanding the Moon's density ratio compared to Earth's helps us comprehend why gravity on the Moon is weaker. In this case, the Moon's mean density is two-thirds that of Earth:

• Density of Moon: \(\frac{2}{3} \times \) Density of Earth

This ratio indicates that the Moon is not only smaller in size but also less dense. A smaller and less dense object will have less gravitational pull than a larger and denser one. The concept of mean (average) density also includes the effect of both the Moon's mass and its volume, influencing its gravitational force exertion capacity.With a lower density, the Moon exerts less gravity despite being composed of similar materials. This density ratio and its calculation are crucial in understanding the overall gravitational influence the Moon has compared to the Earth. It succinctly explains why the Moon's gravitational force is much weaker, allowing for the increased jump height potential for an astronaut.