Problem 56 - University Calculus: Early Transcendentals

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 hits the bottom at 48 m/s.

Step 1: Understanding the Problem

We need to determine the speed of a rock after it has fallen for 30 seconds under moon's gravity, which is given as 1.6 m/s².

Step 2: Identifying the Relevant Formula

For an object in free fall, its velocity after a certain time can be found using the formula: \[ v = g imes t \]where \( v \) is the final velocity, \( g \) is the acceleration due to gravity, and \( t \) is the time.

Step 3: Plugging in the Given Values

Given that \( g = 1.6 \text{ m/s}^2 \) and \( t = 30 \text{ seconds} \), substitute these values into the formula:\[ v = 1.6 \times 30 \]

Step 4: Calculating the Velocity

Multiply the acceleration by the time:\[ v = 1.6 imes 30 = 48 ext{ m/s} \]

Key Concepts

Acceleration due to Gravity

The concept of acceleration due to gravity is fundamental in understanding free fall motion. Gravity is a force that attracts two bodies towards each other. For celestial objects like the moon or Earth, this force causes objects to accelerate when they fall. On Earth, the acceleration due to gravity is approximately 9.8 m/s², but on the moon, it's much less, only 1.6 m/s². This means that objects on the moon fall slower than they would on Earth. In free fall, gravity is the only force acting on an object, making its motion predictable. Knowing the acceleration due to gravity allows us to calculate how fast an object is moving at any given time during its fall. Remember, an object's initial velocity doesn't affect the gravitational acceleration it experiences; gravity does all the work!

• Moon's gravity: 1.6 m/s²
• Earth's gravity: 9.8 m/s²
• Faster fall on Earth due to stronger gravity

Final Velocity Calculation

To determine the speed of a falling object just before it hits the ground, we calculate its final velocity. This is essential in predicting an object's behavior under free fall. The formula used is simple: \[ v = g \times t \]Where:- \( v \) is the final velocity- \( g \) is the acceleration due to gravity- \( t \) is the time the object has been falling.This formula tells us that the final velocity of an object in free fall is directly proportional to the time it has been falling. The longer it falls, the faster it gets. With the moon's gravity of 1.6 m/s², a rock falling for 30 seconds reaches a final velocity:\[ v = 1.6 \times 30 = 48\text{ m/s} \]This velocity calculation helps us understand how fast objects hit the ground under different gravitational conditions. It's straightforward, yet incredibly powerful.

Time of Descent

The time of descent refers to how long an object is in free fall. The longer the time, the more the object accelerates due to gravity. In the moon example, the rock is in free fall for 30 seconds. During this time, the gravitational pull continuously increases its speed until it reaches the calculated final velocity. Factors that affect the time of descent include: - Initial height: Higher altitude leads to longer fall time. - Acceleration due to gravity: Variations in gravitational pull impact the descent time. In problems like our example, knowing the time of descent allows you to calculate important aspects of free fall, such as distance fallen and final velocity. The moon case is interesting because even with a weaker gravitational pull, a significant 30-second fall provides valuable insights into the nature of gravity and motion.

• Longer fall time equals faster speed increase
• Moon's lower gravity affects descent
• Time is crucial for calculating velocity and distance