Question

The acceleration due to gravity on the surface of Mars at the Equator is \(3.699 \mathrm{~m} / \mathrm{s}^{2} .\) How long does it take for a rock dropped from a height of \(1.013 \mathrm{~m}\) to hit the surface?

Short answer

Answer: It takes approximately 0.738 seconds for the rock to hit the surface of Mars at the Equator when dropped from a height of 1.013 meters.

Step-by-step solution

Identify the given values and required variables

We are given the height (h = 1.013 m), and acceleration due to gravity (g = 3.699 m/s^2). The rock's initial velocity (v_initial) will be 0. Our goal is to find the time (t) it takes for the rock to hit the surface.

Equation of motion

We will use the equation of motion that relates distance, initial velocity, time, and acceleration to solve for the time. \(h = v_\text{initial} \cdot t + \frac{1}{2} \cdot g \cdot t^2\)

Substitute the given values

Since the rock is dropped from rest, the initial velocity of the rock (v_initial) is 0. Now substitute the given values into the equation: \(1.013 = 0 \cdot t + \frac{1}{2} \cdot 3.699 \cdot t^2\)

Simplify the equation

We can now simplify the equation to solve for time (t): \(1.013 = 1.8495 \cdot t^2\)

Solve for time (t)

Divide both sides of the equation by 1.8495 to isolate the time squared term: \(t^2 = \frac{1.013}{1.8495}\) Now, take the square root of both sides to find the time: \(t = \sqrt{\frac{1.013}{1.8495}}\)

Calculate the time (t)

Use a calculator to determine the value of time (t): \(t \approx 0.738 \thinspace \mathrm{s}\) So, it takes approximately 0.738 seconds for the rock to hit the surface of Mars at the Equator when dropped from a height of 1.013 meters.

Key concepts

Motion Equations

Understanding motion equations is crucial for solving physics problems involving movement. Motion equations relate various physical quantities like distance, displacement, time, velocity, and acceleration. The most common motion equations are derived from the laws of kinematics and are applicable in situations where the acceleration is constant.

One of the key motion equations used in the exercise is a kinematic equation for uniformly accelerated motion, which is given by: \[h = v_\text{initial} \cdot t + \frac{1}{2} \cdot a \cdot t^2\]In this formula, \(h\) represents the vertical distance (height), \(v_\text{initial}\) is the initial velocity, \(t\) stands for time, and \(a\) denotes acceleration. When an object is dropped, its initial velocity is zero, simplifying the equation. By rearranging and solving this equation, one can find the time taken for an object to hit the ground when dropped from a certain height with known acceleration, such as gravity on Mars.

Free Fall

Free fall is the motion of an object under the influence of gravitational force only. When a rock is dropped from a height, as in the exercise, it's an example of free fall. The only force acting on the object is gravity, causing it to accelerate downwards. The acceleration due to gravity on Earth is approximately \(9.81 \mathrm{\,m/s^2}\), but it can vary on different planets or celestial bodies.

For the exercise problem, knowing that Mars has a weaker gravitational field with an acceleration due to gravity of \(3.699 \mathrm{\,m/s^2}\) allows students to analyze free fall in different gravitational contexts. In free fall, no initial velocity and no air resistance mean the motion equations for free fall become much simpler to apply.

Kinematics

Kinematics is the branch of mechanics that deals with pure motion, without considering the forces that cause it. It is fundamental for understanding how objects move in space and time. The core concepts of kinematics include velocity, acceleration, displacement, and time. These quantities are related through kinematic equations, which can describe the motion of objects with constant acceleration.

In the context of the given exercise, kinematics allows us to deduce how the rock falls and to predict how long it takes to reach the ground by using the appropriate motion equation. To solve kinematic problems, clear visualization of the scenario and careful selection of equations are essential. Students are encouraged to sketch the problem, label all known and unknown quantities, and choose the correct motion equation to apply.

Physics Problem Solving

Physics problem solving is a systematic approach to understanding and solving physics-based scenarios. A well-defined problem-solving strategy involves identifying known and unknown variables, visualizing the problem, choosing the right formula, substituting with the correct values, and performing the calculations accurately.

In the exercise, the step-by-step solution follows this approach: specifying the knowns (height and acceleration), recognizing the unknown (time), selecting an appropriate motion equation, substituting the values, and ultimately solving for the desired quantity. It's important for students to practice this methodical process as it enhances their problem-solving skills and helps them tackle a wide range of physics problems efficiently.