Question

The planet Mercury has a mass that is \(5 \%\) of that of Earth, and its gravitational acceleration is \(g_{\text {mercury }}=3.7 \mathrm{~m} / \mathrm{s}^{2}\) a) How long does it take for a rock that is dropped from a height of \(1.75 \mathrm{~m}\) to hit the ground on Mercury? b) How does this time compare to the time it takes the same rock to reach the ground on Earth, if dropped from the same height? c) From what height would you have to drop the rock on Earth so that the fall- time on both planets is the same?

Short answer

Based on the above solution, if a rock falls from a height of 1.75 meters on Mercury, it takes approximately 0.974 seconds for it to hit the ground, while on Earth, it takes approximately 0.598 seconds to fall from the same height. In order to have the same fall-time on both Mercury and Earth, the rock needs to be dropped from a height of approximately 4.625 meters on Earth.

Step-by-step solution

Determine the kinematic equation for the fall-time

We will use the standard kinematic equation for free-fall motion to calculate the fall time. The equation is: \(h = \frac{1}{2}gt^2\). Here, \(h\) is the height, \(g\) is the gravitational acceleration, and \(t\) is the time.

Plug in the values for Mercury

We will plug in the values for Mercury: \(h = 1.75~m\) and \(g = 3.7~m/s^2\). By rearranging the equation we get: \(t = \sqrt{\frac{2h}{g}}\).

Compute the fall-time on Mercury

Now, we calculate the fall-time for the rock on Mercury: \(t = \sqrt{\frac{2(1.75)}{3.7}} \approx 0.974~s\). b) Time comparison with Earth:

Determine Earth's gravitational acceleration

The gravitational acceleration on Earth is: \(g_{\text {earth }}=9.81~m /s^{2}\).

Compute the fall-time on Earth

Using the equation from Step 2, we plug in the values for Earth: \(h = 1.75~m\) and \(g_{\text {earth }} = 9.81~m/s^{2}\). Then, we calculate the fall-time for the rock on Earth: \(t_{\text {earth }}=\sqrt{\frac{2(1.75)}{9.81}}\approx 0.598~s\).

Compare the fall-times on both planets

It takes approximately 0.974 seconds for the rock to hit the ground on Mercury and 0.598 seconds on Earth. The rock takes longer to fall on Mercury due to its weaker gravitational force. c) Same fall-time on both planets:

Set up an equation with equal fall-times

Using the equation from step 2 for both planets, we have: \(\sqrt{\frac{2h_{\text {earth }}}{g_{\text {earth }}}}=\sqrt{\frac{2h_{\text {mercury }}}{g_{\text {mercury }}}}\) Since the fall-time should be the same on both planets, we can equate them: \(\frac{2h_{\text {earth }}}{g_{\text {earth }}}=\frac{2h_{\text {mercury }}}{g_{\text {mercury }}}\)

Solve for the height on Earth

Rearrange the equation to find the height on Earth: \(h_{\text {earth }}=\frac{g_{\text {earth }}}{g_{\text {mercury }}}h_{\text {mercury}}\)

Plug in the values for Earth and Mercury

We plug in the values for both planets and calculate the height on Earth: \(h_{\text {earth }}=\frac{9.81}{3.7}(1.75) \approx 4.625~m\)

Conclusion

The rock has to be dropped from a height of approximately 4.625 meters on Earth in order for the fall-time to be the same as on Mercury.

Key concepts

Free-fall Motion

When an object is dropped from a certain height without any initial velocity, that motion is termed as free-fall. This means it is only under the influence of gravitational force, pulling it toward the surface of a planet.
Gravitational acceleration differs depending on which planet you're on. For example, Mercury has a gravitational acceleration of 3.7 m/s², while Earth's is 9.81 m/s². This difference in gravitational force greatly affects how quickly an object accelerates as it falls.
As a result, a rock will fall more slowly on Mercury compared to Earth due to the weaker gravitational pull. It's crucial to remember that in free-fall, air resistance is negligible, allowing us to solely focus on gravity's effects.

Kinematic Equations

Kinematic equations are a set of formulas that relate the five key variables of motion: displacement (\(h\)), initial velocity (\(v_i\)), final velocity (\(v_f\)), acceleration (\(a\)), and time (\(t\)). They are particularly useful in solving problems involving constant acceleration, such as objects in free-fall motion.
In our exercise, the relevant kinematic equation is:\[h = \frac{1}{2}gt^2\]Here, \(h\) is the height from which the object is dropped, \(g\) is the acceleration due to gravity, and \(t\) is the time it takes to reach the ground. This equation assumes the initial velocity is zero, which is true for free-fall drop situations.
By rearranging this formula, you can solve for time (\(t\)) as follows:\[t = \sqrt{\frac{2h}{g}}\]This allows us to find how long it takes an object to fall from a given height by substituting the values of \(h\) and \(g\) for the respective planet.

Earth and Mercury Comparison

Understanding how gravitational acceleration affects free-fall can be illuminated through a comparison of Earth and Mercury. Mercury's gravitational acceleration is significantly less than Earth's at 3.7 m/s² compared to 9.81 m/s², meaning objects fall slower on Mercury.
To see this difference in action, let's consider the time it takes for a rock to fall 1.75 meters. On Mercury, it takes approximately 0.974 seconds, whereas on Earth, the same rock only takes about 0.598 seconds. This shows that due to Mercury's weaker gravity, fall times are significantly longer.
Another interesting calculation is finding from what height you must drop the rock on Earth to match Mercury's fall-time. Recalculating the height using the kinematic equation, it comes out to approximately 4.625 meters on Earth. Therefore, reaching the same fall-time on both planets requires considering their distinctive gravitational influences.