Problem 50 - College Physics

Question

One way to measure \(g\) on another planet or moon by remote sensing is to measure how long it takes an object to fall a given distance. A lander vehicle on a distant planet records the fact that it takes \(3.17 \mathrm{~s}\) for a ball to fall freely \(11.26 \mathrm{~m},\) starting from rest. (a) What is the acceleration due to gravity on that planet? Express your answer in \(\mathrm{m} / \mathrm{s}^{2}\) and in earth \(g^{\prime} \mathrm{s}\). (b) How fast is the ball moving just as it lands?

Step-by-Step Solution

Answer

(a) \( g \approx 2.238 \ \mathrm{m/s^2} \), (b) Final velocity \( \approx 7.09 \ \mathrm{m/s} \).

Step 1: Identify known values

We have the initial velocity \( v_0 = 0 \ \mathrm{m/s} \), the time \( t = 3.17 \ \mathrm{s} \), and the distance \( s = 11.26 \ \mathrm{m} \). We seek the acceleration \( g \).

Step 2: Use the equation for distance

The equation for distance in free fall is \( s = v_0 t + \frac{1}{2} g t^2 \). Substitute the known values: \( 11.26 = 0 \cdot 3.17 + \frac{1}{2} g (3.17)^2 \).

Step 3: Solve for acceleration \( g \)

Simplify the equation: \( 11.26 = \frac{1}{2} g (10.0489) \). Hence, \( g = \frac{11.26 \times 2}{10.0489} \). Calculating gives \( g \approx 2.238 \ \mathrm{m/s^2} \).

Step 4: Convert to earth \( g \)

The acceleration due to gravity on Earth is about \( 9.8 \ \mathrm{m/s^2} \). Hence, the planetary \( g \) in Earth \( g \)s is \( \frac{2.238}{9.8} \approx 0.228 \ g \).

Step 5: Calculate final velocity when ball lands

Using \( v = v_0 + gt \), substitute \( v_0 = 0 \), \( g = 2.238 \), and \( t = 3.17 \): \( v = 0 + 2.238 \times 3.17 \). Simplifying gives \( v \approx 7.09 \ \mathrm{m/s} \).

Key Concepts

Acceleration Due to Gravity

The acceleration due to gravity, often denoted by the symbol \(g\), is a measure of the gravitational force exerted on an object at the surface of a planet or moon. It is essential for understanding how objects behave under the influence of gravity. On Earth, the standard value is approximately \(9.8 \, \mathrm{m/s^2}\). This value signifies that, excluding air resistance, any object in free fall will accelerate downwards at this rate.

In the context of the given exercise, determining the value of \(g\) on another planet involves measuring how long it takes for an object to fall a specific distance. Using the formula for motion in physics, which is derived from Newton's second law, one can calculate the gravitational acceleration by examining this free-fall time. By employing the equation \( s = v_0 t + \frac{1}{2} g t^2 \), where \(v_0\) is the initial velocity (0 in this case), you can derive \(g\) to understand the planet's gravitational pull.

Planetary Science

Planetary science is a branch of astronomy that involves the study of planets, moons, and planetary systems, especially within our solar system. Understanding gravity on other planets is crucial for these studies because it affects everything from the planet's atmosphere to the way space probes land on its surface.

The process of measuring gravity on a distant planet can be done through remote sensing or lander missions. This is essential for a multitude of reasons:

• Determining surface conditions that might affect future explorations or colonization.
• Understanding the planet's internal structure based on its gravitational footprint.
• Assessing potential resources in terms of minerals and materials.

Through these measurements, scientists can make educated predictions and considerations about the habitability and underlying processes happening on the surface and within the planet.

Free Fall

Free fall refers to the condition where an object is moving under the influence of gravitational force alone, without any resistance or support. When a ball is dropped from a height on a planet, whether Earth or elsewhere, and we assume no air resistance, it's said to be in free fall.

The concept of free fall is central to understanding acceleration due to gravity. The formula \( s = v_0 t + \frac{1}{2} g t^2 \) describes how far an object falls within a certain time, starting from rest. In the exercise scenario, the time taken and distance fallen were known, allowing for the computation of \(g\) by rearranging and solving the equation. This shows free fall isn't just a concept but a practical tool for measuring gravitational forces.

Kinematics

Kinematics is the branch of physics concerned with the motion of objects without considering forces. It deals with parameters like velocity, acceleration, displacement, and time, and uses these to determine unknowns within motion scenarios.

In the example problem, kinematics allows us to calculate both the acceleration due to gravity and the final velocity of the ball at impact. Starting with the kinematic equation for distance \( s = v_0 t + \frac{1}{2} g t^2 \) and solving for \(g\) gives insight into the planet's gravity. Furthermore, using the equation \( v = v_0 + gt \) helps us find the velocity of the ball just before landing.

Kinematics not only provides a method for solving such exercises but also plays a crucial role in designing experiments and interpreting results in both terrestrial and extraterrestrial environments.