Problem 54 - Thomas Calculus

Question

Free fall on the moon On our moon, the acceleration of gravity is 1.6 \(\mathrm{m} / \mathrm{sec}^{2} .\) If a rock is dropped into a crevasse, how fast will it be going just before it hits bottom 30 sec later?

Step-by-Step Solution

Answer

The rock will be traveling at 48 m/s just before it hits the bottom.

Step 1: Understand the Problem

We need to calculate the velocity of a rock just before it hits the bottom of a crevasse on the moon after being in free fall for 30 seconds. Given that the acceleration due to gravity on the moon is \(1.6\,\text{m/s}^2\).

Step 2: Use the Basic Kinematic Equation for Final Velocity

The basic kinematic equation for velocity under constant acceleration is \(v = u + at\), where \(v\) is the final velocity, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time. Since the rock is dropped (not thrown), \(u = 0\).

Step 3: Substitute Known Values into the Equation

Substitute \(u = 0\), \(a = 1.6\,\text{m/s}^2\), and \(t = 30\,\text{s}\) into the equation: \[v = 0 + (1.6\,\text{m/s}^2)(30\,\text{s})\]

Step 4: Calculate the Final Velocity

Calculate the final velocity: \[v = 1.6 \times 30 = 48\,\text{m/s}\]The velocity just before the rock hits the bottom is \(48\,\text{m/s}\).

Key Concepts

Understanding Free Fall

Free fall is a special kind of motion in kinematics. It occurs when an object moves under the influence of gravity alone, without any other forces acting on it. The rock in our exercise is undergoing free fall on the moon. Here are some key points to remember:

• Free fall always involves gravitational acceleration.
• The object does not experience any support or resistance—no air resistance in this simple scenario.
• Initial velocity can vary: in this case, since the rock is dropped, the initial velocity is zero.

Let's visualize: imagine the rock being released from rest on the barren lunar surface. As it falls, the only force acting upon it is the moon's gravitational pull. This is what sets free fall apart from other kinds of motion.

Acceleration Due to Gravity

Acceleration due to gravity is a constant force that acts upon all objects in a planet's vicinity. On Earth, this value is approximately 9.81 m/s² but is different on the moon, where it is 1.6 m/s². Here's why this matters:

• Gravity affects how fast an object speeds up as it falls.
• On the moon, less gravity means objects fall more slowly than on Earth.
• All objects, regardless of mass, experience the same acceleration if other forces are negligible.

In the exercise, the rock's acceleration is solely determined by the moon's gravitational pull—1.6 m/s². This lower planetary gravity affects how quickly the rock reaches the speed just before impact. It shows how different celestial bodies influence motion with their unique gravitational force.

Calculating Final Velocity

The final velocity of an object in free fall can be calculated using the kinematic equation: \[ v = u + at \] Here, \(u\) is the initial velocity, \(a\) is the acceleration, and \(t\) is the time for which the object has been falling. In our exercise:

• Since the rock is initially dropped, \(u = 0\).
• The acceleration \(a\) is 1.6 m/s².
• The time \(t\) it falls is given as 30 seconds.

Substituting these values into the equation gives:\[ v = 0 + (1.6 \, \text{m/s}^2)(30) = 48 \, \text{m/s} \]So, the rock's final velocity, just before it hits the bottom, is 48 m/s. This calculation demonstrates kinematics in action, using an equation to determine how speed changes over time under constant acceleration.