Problem 52 - College Physics

Question

Astronauts on our moon must function with an acceleration due to gravity of \(1.62 g .\) (a) If an astronaut can throw a certain wrench \(12.0 \mathrm{~m}\) vertically upward on earth, how high could he throw it on our moon if he gives it the same starting speed in both places? (b) How much longer would it be in motion (going up and coming down) on the moon than on earth?

Step-by-Step Solution

Answer

(a) 72.8 m on the moon. (b) 16.22 s longer on the moon.

Step 1: Determine the initial velocity of the wrench on Earth

Use the equation for vertical motion, where the final velocity is zero at the maximum height: \ \ \(0 = v_i^2 - 2g_\text{earth} h_\text{e} \). \ \ Here, \(g_\text{earth} = 9.8 \, \text{m/s}^2\) and \(h_\text{e} = 12.0 \, \text{m}\). \ \ Solve for \(v_i\): \ \ \( v_i^2 = 2 \cdot 9.8 \cdot 12.0 \) \ \ \( v_i^2 = 235.2 \) \ \ \( v_i = \sqrt{235.2} \approx 15.34 \, \text{m/s} \).

Step 2: Calculate the maximum height on the moon

Using the same initial speed on the moon, apply the same equation for vertical motion: \ \ \(0 = v_i^2 - 2g_\text{moon} h_\text{m} \), \ \ where \(g_\text{moon} = 1.62 \times 9.8 = 15.876 \, \text{m/s}^2\). \ \ Solve for \(h_\text{m}\): \ \ \(0 = 235.2 - 2 \cdot 1.62 \cdot 9.8 \cdot h_\text{m} \). \ \ \( h_\text{m} = \frac{235.2}{2 \cdot 1.62 \cdot 9.8} \approx 72.8 \, \text{m} \).

Step 3: Calculate time of flight on Earth

Use the time of flight formula for the upward and downward journey: \ \ \(t = \frac{2v_i}{g_\text{earth}} \). \ \ \(t = \frac{2 \times 15.34}{9.8} = 3.13 \, \text{s}\).

Step 4: Calculate time of flight on the moon

Using the same formula, but with moon's gravity: \ \ \(t = \frac{2v_i}{g_\text{moon}} \), \ \ where \(g_\text{moon} = 15.876 \, \text{m/s}^2\). \ \ \(t = \frac{2 \times 15.34}{1.62 \times 9.8} \approx 19.35 \, \text{s}\).

Step 5: Calculate the difference in time

Subtract the time of flight on Earth from the time on the moon: \ \ \(\Delta t = t_\text{moon} - t_\text{earth} = 19.35 - 3.13 \approx 16.22 \, \text{s}\).

Key Concepts

Kinematics

Kinematics is a branch of physics that explores the motion of objects. It focuses on parameters such as velocity, acceleration, and displacement, without considering the forces causing the motion.
In the context of projectile motion, kinematics helps to describe the path (trajectory) and behavior of a moving object under the influence of gravity, typically assuming a single direction in motion, such as vertically upward or downward.

Key aspects of kinematics include:

• Displacement: Change in position of the object. In our example, this is the height the wrench travels.
• Velocity: The speed of the object in a specified direction. For projectiles, this changes due to gravity.
• Acceleration: The rate of change of velocity. For our exercise, it is influenced by gravity, which changes between the Earth and the Moon.

By applying kinematic equations, we can predict how an object behaves when subjected to known forces and conditions, aiding in solving problems like determining maximum height or time of flight.

Moon Gravity

Gravity on the Moon is significantly weaker than on Earth. It is approximately one-sixth of Earth's gravity. This means that objects on the Moon will experience less gravitational force pulling them down than they do on Earth.

On Earth, standard gravity is considered to be 9.8 meters per second squared (m/s²). On the Moon, this is reduced to about 1.62 times Earth's gravity when recalculated, or 15.876 m/s², for practical purposes.

• Impact of Weaker Gravity: Objects can be thrown or moved over much greater distances with the same effort as on Earth.
• Implications for Astronauts: Movement becomes easier, but controlling mobility and stability can be challenging due to longer airborne time and altered trajectory curves.

Studying Moon's gravity is crucial for preparing astronauts and equipment for missions beyond Earth, ensuring they can operate effectively in reduced gravity environments.

Initial Velocity

Initial velocity is the speed at which an object is thrown or launched. It establishes the starting point for analyzing the object's motion under known conditions.
In projectile motion problems, the initial velocity can determine several subsequent factors, such as the maximum height reached or total time of flight. It is a critical factor used in kinematic equations when evaluating projectile behaviors.

• Calculation on Earth: The initial velocity was calculated to be approximately 15.34 m/s, based on the information given about how high the wrench can be thrown upwards under Earth's gravity.
• Consistency Across Environments: Despite differing gravitational forces, the initial velocity in this exercise remains constant between the Earth and Moon, as it depends solely on the power exerted by the astronaut.

Understanding initial velocity is vital when planning the launch or motion paths of various objects, helping predict future motion accurately.

Time of Flight

Time of flight refers to the total time an object spends in the air from the moment it is launched until it lands back. This duration is influenced by factors such as initial speed, launch angle, and gravitational acceleration,
which can differ significantly between the Earth and the Moon.
In our exercise:

• On Earth: The time of flight for the wrench was calculated to be about 3.13 seconds. This represents both the ascent and the descent time.
• On the Moon: In comparison, the lesser gravitational pull extended the time of flight to approximately 19.35 seconds, highlighting the prominent impact of reduced gravity on motion duration.

• Impact on Calculations: Knowing the time of flight allows one to calculate other projectile parameters, such as range or maximum height.

Time of flight is especially important in space missions, where accuracy in movement and timing is critical for safe and successful operations.