Understanding how gravity affects jump height is key to solving this problem. When a person jumps, they exert a force to propel themselves upwards, gaining an initial vertical velocity. The maximum height they reach is determined by this initial velocity and the acceleration due to gravity pulling them back down.

Let's consider the physics involved in a vertical jump. Ignoring air resistance, the motion is governed by constant acceleration, which is the acceleration due to gravity, \(g\). The maximum height (\(h\)) is reached when the upward velocity becomes zero. Using the kinematic equation \(v^2 = u^2 + 2as\), where:

• \(v\) is the final velocity (0 at the peak height)
• \(u\) is the initial velocity (let's call it \(v_0\))
• \(a\) is the acceleration (\(-g\))
• \(s\) is the displacement (\(h\))

We get:

\(0^2 = v_0^2 + 2(-g)h\)

\(0 = v_0^2 - 2gh\)

\(2gh = v_0^2\)

From this, the maximum height \(h\) is given by the formula:

\(h = \frac{v_0^2}{2g}\)

Now, we are comparing the jump height on Earth and on the Moon. The question states that a man can jump 1.5 m high on the Earth. This jump is achieved by the man exerting his muscles to produce an initial upward velocity. It's reasonable to assume that the man can produce the same initial velocity (\(v_0\)) with the same effort on both Earth and the Moon, as his physical capacity doesn't change, only the gravity does.

Let \(h_e\) be the maximum jump height on Earth and \(g_e\) be the acceleration due to gravity on Earth. Let \(h_m\) be the maximum jump height on the Moon and \(g_m\) be the acceleration due to gravity on the Moon.

Using the formula \(h = \frac{v_0^2}{2g}\):

• On Earth: \(h_e = \frac{v_0^2}{2g_e}\)
• On the Moon: \(h_m = \frac{v_0^2}{2g_m}\)

We are given that the value of \(g\) on the moon is 1/6 th of the value of \(g\) on the earth. Mathematically, this is expressed as:

\(g_m = \frac{1}{6} g_e\)

We want to find the jump height on the Moon (\(h_m\)). We can set up a ratio of the jump heights on the Moon and Earth:

\(\frac{h_m}{h_e} = \frac{\frac{v_0^2}{2g_m}}{\frac{v_0^2}{2g_e}}\)

The term \(\frac{v_0^2}{2}\) is the same for both Earth and Moon, assuming the initial velocity \(v_0\) is the same. So, it cancels out:

\(\frac{h_m}{h_e} = \frac{\frac{1}{g_m}}{\frac{1}{g_e}} = \frac{g_e}{g_m}\)

Now, substitute the relationship between \(g_m\) and \(g_e\):

\(\frac{h_m}{h_e} = \frac{g_e}{\frac{1}{6} g_e}\)

\(\frac{h_m}{h_e} = \frac{1}{\frac{1}{6}}\)

\(\frac{h_m}{h_e} = 6\)

This tells us that the jump height on the Moon is 6 times the jump height on Earth, provided the initial velocity is the same.

We are given that the man can jump \(h_e = 1.5\) m high on the Earth.

Using the relationship \(h_m = 6 \times h_e\):

\(h_m = 6 \times 1.5\text{ m}\)

\(h_m = 9.0\text{ m}\)

So, the man can jump up to a height of 9 m on the Moon.

Comparing this result with the given options:

• 4.5 m
• 9 m
• 6 m
• 7.5 m

The calculated height of 9 m matches one of the options.

Revision Table: Jump Height and Gravity

Additional Information on Gravity and Jump Height

The acceleration due to gravity (\(g\)) is a measure of the force per unit mass exerted by a gravitational field. It depends on the mass and radius of the celestial body. The formula for gravity on the surface is approximately:

\(g = \frac{GM}{R^2}\)

• \(G\) is the gravitational constant
• \(M\) is the mass of the celestial body
• \(R\) is the radius of the celestial body

The Moon has less mass and a smaller radius than Earth, but the effect of the lower mass dominates, resulting in a significantly lower surface gravity, which is about 1/6th of Earth's gravity.

Because jump height is inversely proportional to the acceleration due to gravity (\(h \propto \frac{1}{g}\)), a lower \(g\) means a higher jump is possible with the same initial velocity. This is why astronauts on the Moon could jump much higher than they can on Earth, even in their bulky suits.

Factors not considered in this simplified model include air resistance (negligible on the Moon) and potential differences in muscle efficiency in different gravitational environments, but the primary factor affecting jump height difference between Earth and Moon for the same person is the change in gravity.