Problem 79 - Physics for Scientists and Engineers with Modern Physics

Question

Apollo astronauts took a "nine iron" to the Moon and hit a golf ball about \(180 \mathrm{~m}\). Assuming that the swing, launch angle, and so on, were the same as on Earth where the same astronaut could hit it only \(32 \mathrm{~m}\), estimate the acceleration due to gravity on the surface of the Moon. (We neglect air resistance in both cases, but on the Moon there is none.)

Step-by-Step Solution

Answer

The acceleration due to gravity on the Moon is approximately \(1.74 \text{ m/s}^2\).

Step 1: Understand the Problem

We need to compare the range of a golf ball hit on the Moon versus on Earth, given that the launch conditions (like angle and swing) are the same. By using this comparison, we'll estimate the Moon's gravitational acceleration.

Step 2: Recall the Range Formula for Projectile Motion

The range of a projectile is given by the formula: \[ R = \frac{v^2 \sin(2\theta)}{g} \]where \( R \) is the range, \( v \) is the launch speed, \( \theta \) is the launch angle, and \( g \) is the acceleration due to gravity. The sine function \( \sin(2\theta) \) is the same for both planets as the launch angle is identical.

Step 3: Set up the Ratio of the Ranges

Since the launch conditions are the same, we set up a ratio comparing the ranges on Moon \( (R_m) \) and Earth \( (R_e) \):\[ \frac{R_m}{R_e} = \frac{g_e}{g_m} \]Inserting the given data: \( R_m = 180 \text{ m} \), \( R_e = 32 \text{ m} \), and Earth \( g_e = 9.8 \text{ m/s}^2 \).

Step 4: Solve for the Moon's Gravity

Using the ratio:\[ \frac{180}{32} = \frac{9.8}{g_m} \]Solving for \( g_m \):\[ g_m = \frac{9.8 \times 32}{180} \approx 1.74 \text{ m/s}^2 \]

Step 5: Verification

Check the calculation:\[ g_m \approx \frac{313.6}{180} = 1.7422 \text{ m/s}^2 \]This confirms the solution. The gravitational acceleration on the moon is accurate to around \(1.74 \text{ m/s}^2\).

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object thrown into the air, influenced only by the force of gravity. Examples include throwing a ball or shooting an arrow. In these scenarios, the object travels along a curved path due to the initial velocity and the gravitational pull acting on it.

Understanding projectile motion involves recognizing two important components: horizontal and vertical motion. These components are independent of one another. The horizontal motion is constant since air resistance is negligible (especially on the Moon where there is no air!), while the vertical motion is affected by gravitational acceleration.

When discussing projectile motion, remember these key points:

• The initial velocity and angle of launch determine how far and high the object will travel.
• Gravity only affects the vertical component of motion.
• Air resistance, if present, would slow down the object, but in space such as on the Moon, it's important to note that this isn't a factor.

Understanding these concepts helps us solve problems involving distances or heights achieved by moving objects.

Gravitational Acceleration on the Moon

Gravitational acceleration is the rate at which an object speeds up as it falls due to gravity. On the Earth, this rate is roughly 9.8 m/s², but it differs on the Moon because of its smaller mass.

When the Apollo astronaut hit the golf ball, it traveled a greater distance on the Moon compared to the Earth. This happens because the Moon's gravity is weaker, thus the gravitational acceleration is lower.

In this specific problem, gravitational acceleration on the Moon is calculated to be approximately 1.74 m/s². We derived this using the ratio of the golf ball’s ranges on the Moon and Earth, and knowing Earth’s gravitational acceleration. With lower gravity, objects weigh less and fall more slowly compared to Earth. That's why your golf ball would travel farther, given the same swing!

This difference in gravitational pull is crucial when planning missions and activities on the Moon, as it directly impacts how objects behave.

Range Formula in Physics

The range formula is a vital tool in physics used to calculate the distance a projectile will travel. It's expressed as:\[ R = \frac{v^2 \sin(2\theta)}{g} \]where:

• \( R \) is the range, or the distance traveled.
• \( v \) is the launch speed.
• \( \theta \) is the launch angle.
• \( g \) is the acceleration due to gravity.

This formula is particularly useful because it separates the effects of speed and angle from gravity. The \(\sin(2\theta)\) part becomes crucial in maximizing distance, as the angle directly affects the projectile's trajectory.

In the context of the exercise, the range formula allowed us to set up a ratio between the ranges achieved on Earth and the Moon, given identical launch speeds and angles. This comparison and manipulation of the formula showed how different gravitational fields impact projectile motion, and in our case, helped estimate the gravitational acceleration on the Moon.