Problem 18 - An Introduction to Physical Science

Question

A spaceship hovering over the surface of Mars drops an object from a height of 30 m. How much longer does it take to reach the surface than if dropped from the same height on Earth? Neglect air resistance in both cases. [The acceleration due to gravity on Mars is \(33 \%\) of that on Earth, \(\left.g_{\text {mars }}=(0.33) g .\right]\)

Step-by-Step Solution

Answer

It takes 1.82 seconds longer for the object to reach the surface on Mars.

Step 1: Define Known Variables

First, find out the known variables and constants. The height from which the object is dropped is 30 m. The gravitational acceleration on Earth is usually taken as \( g = 9.81 \, \text{m/s}^2 \). On Mars, it is \( g_{\text{mars}} = 0.33g = 0.33 \times 9.81 \, \text{m/s}^2 \).

Step 2: Derive Time Formula on Mars

For an object in free fall, time to reach the surface can be calculated by the formula: \( t = \sqrt{\frac{2h}{g}} \). Applying this on Mars: \( t_{\text{mars}} = \sqrt{\frac{2 \times 30}{0.33 \times 9.81}} \).

Step 3: Derive Time Formula on Earth

Similarly, for Earth, we use the same formula: \( t_{\text{earth}} = \sqrt{\frac{2 \times 30}{9.81}} \).

Step 4: Calculate Time on Mars

Now compute the time it takes for the object to fall on Mars: \( t_{\text{mars}} = \sqrt{\frac{60}{3.24}} \approx 4.29 \, \text{s} \).

Step 5: Calculate Time on Earth

Next, compute the time for Earth: \( t_{\text{earth}} = \sqrt{\frac{60}{9.81}} \approx 2.47 \, \text{s} \).

Step 6: Comparison of Times

To find how much longer it takes on Mars: \( \Delta t = t_{\text{mars}} - t_{\text{earth}} = 4.29 - 2.47 = 1.82 \, \text{s} \).

Key Concepts

Free Fall

Free fall is an intriguing concept in physics. Imagine dropping an object, with no initial velocity, and letting it fall solely under the influence of gravity. This is what we call free fall. In this condition, the only force acting on the object is the gravitational pull of the planet it is on. As it falls, it accelerates downwards with a constant force. We say the object is in free fall because we ignore other forces that might act on it, such as air resistance. This simplifies calculations and helps us focus on the role of gravity.In free fall, we can calculate the time it takes for an object to hit the ground using the formula:

• \( t = \sqrt{\frac{2h}{g}}\)

Where:

• \(t\) is the time in seconds,
• \(h\) is the height in meters, and
• \(g\) is the gravitational acceleration in meters per second squared.

The beauty of this formula is that it can be applied to any situation on any planet, as long as you know the height and the specific gravitational acceleration.

Mars

Mars is known as the 'Red Planet' and is the fourth planet from the Sun in our solar system. One interesting fact about Mars is its weaker gravitational force compared to Earth. When thinking about dropping an object on Mars, especially in terms of free fall, this means objects fall more slowly.Mars' gravitational acceleration is significantly less than Earth's; it is only about 33% of Earth's gravity. This makes calculations quite interesting, as the falling time of an object increases due to the lower force pulling it down. For our exercise, the acceleration on Mars is given by:

• \( g_{\text{mars}} = 0.33 \times 9.81 \approx 3.24 \, \text{m/s}^2 \)

This interesting fact about Mars changes how long it takes for a dropped object to reach the ground compared to how it would behave on Earth. Thus, understanding gravity on Mars provides insights into the unique conditions experienced on space missions or even potential future habitats.

Earth

Earth, the third planet from the Sun, has a rich variety of environments due to its gravitational pull. When objects fall to Earth under this force, they experience a gravitational acceleration of approximately 9.81 meters per second squared. This is a crucial figure in physics, as it determines how quickly objects fall to the surface in free fall.For example, when dropping an object from a height of 30 meters, as in our exercise, the time it would take to reach the ground can be calculated using Earth's gravitational acceleration in the free fall formula:

• \( t_{\text{earth}} = \sqrt{\frac{2 \times 30}{9.81}} \approx 2.47 \, \text{s} \)

Understanding Earth's gravitational acceleration is fundamental to many practical applications, not just in science and engineering but also in our daily lives, such as understanding the physics of sports or designing safe structures.

Gravitational Acceleration

Gravitational acceleration is a key concept that varies depending on where you are in the universe. It represents how strong the gravitational pull is on an object. This is important because it affects the motion of any object in free fall. Every planet has a different gravitational acceleration. For instance, Earth's is about 9.81 m/s². This is considered a standard value in physics for conducting experiments and performing calculations. On Mars, the value is much lower, approximately 3.24 m/s². Gravitational acceleration can significantly influence the dynamics of movements:

• It dictates how fast an object speeds up as it falls.
• It varies on different celestial bodies due to their mass and size. Larger bodies tend to have greater gravitational acceleration, whereas smaller ones have less.
• In scientific explorations, knowing the correct gravitational acceleration of a location helps predict motion and behavior of objects.

It's fascinating how the universe operates with such precision, and understanding these forces deepens our connection to the cosmos and enhances our ability to explore and utilize space.