Problem 55 - College Physics

Question

An astronaut on the Moon fires a projectile from a launcher on a level surface so as to get the maximum range. If the launcher gives the projectile a muzzle velocity of \(25 \mathrm{~m} / \mathrm{s},\) what is the range of the projectile? [Hint: The acceleration due to gravity on the Moon is only onesixth of that on the Earth.]

Step-by-Step Solution

Answer

The range of the projectile on the Moon is approximately 383.44 meters.

Step 1: Understanding the Problem

We need to find the range of a projectile launched on the Moon with a velocity of \(25 \mathrm{~m/s}\). The acceleration due to gravity on the Moon is \(\frac{1}{6}\) of that on Earth, which is approximately \(9.8 \mathrm{~m/s^2}\). Calculate the Moon's gravity \(g_{moon}\) as \(\frac{9.8}{6}\mathrm{~m/s^2}\).

Step 2: Identify Formula for Maximum Range

For maximum range, the projectile must be launched at a 45-degree angle. The formula for the range \(R\) of a projectile under gravity \(g\) with an initial velocity \(v\) is given by: \[ R = \frac{v^2 \sin(2\theta)}{g} \]Since \(\theta = 45^\circ\), \( \sin(90^\circ) = 1\) and the formula simplifies to:\[ R = \frac{v^2}{g} \]

Step 3: Calculate Moon's Gravity

The Earth's gravitational acceleration is \(9.8 \mathrm{~m/s^2}\), so the Moon's gravitational acceleration is:\[ g_{moon} = \frac{9.8}{6} = 1.63 \mathrm{~m/s^2} \]

Step 4: Apply the Range Formula

Substitute \(v = 25 \mathrm{~m/s}\) and \(g_{moon} = 1.63 \mathrm{~m/s^2}\) into the range formula:\[ R = \frac{(25)^2}{1.63} \]

Step 5: Calculate the Range

Compute the value:\[ R = \frac{625}{1.63} = 383.44 \mathrm{~m} \]This is the range of the projectile on the Moon.

Key Concepts

Moon Gravity

Gravity is a force that attracts two bodies towards each other. On Earth, this force gives weight to physical objects and pulls them towards the ground. However, on the Moon, gravity is much weaker. Moon gravity is only about one-sixth of Earth's gravity. This means that if you weigh 60 kg on Earth, you would weigh only 10 kg on the Moon. This lower gravity affects how objects move. For projectiles, the weaker Moon gravity means they can travel much further before touching the ground. So, if you launch a projectile on the Moon, it will have a significantly larger range compared to the same launch on Earth. Understanding the correct value of Moon's gravity, which is approximately 1.63 m/s², is critical in calculating projectile motion on the Moon.

Maximum Range

The maximum range of a projectile is the farthest horizontal distance it can travel when launched. The key to determining this maximum distance lies in the launch angle. When a projectile is launched at a 45-degree angle, it covers the maximum distance possible because it balances vertical and horizontal motion perfectly. The formula to calculate the range is given by \( R = \frac{v^2 \sin(2\theta)}{g} \). For maximum range, \( \theta \) should be 45°. This simplifies the formula to \( R = \frac{v^2}{g} \), where \( v \) is the muzzle velocity and \( g \) is the gravitational acceleration. On the Moon, calculating the maximum range is straightforward once you know the muzzle velocity and Moon gravity.

Muzzle Velocity

Muzzle velocity is the speed with which a projectile leaves the launching device. It is crucial in determining how far and how fast a projectile will travel. In the context of projectile motion, a higher muzzle velocity will generally lead to a greater range. In our exercise, the projectile had a muzzle velocity of 25 m/s. This high velocity, paired with the Moon's low gravity, contributes to the impressive range of the projectile. The formula for calculating range shows us that the range is directly proportional to the square of the muzzle velocity \( (v^2) \), emphasizing its importance. Understanding muzzle velocity is not only important for solving projectile motion problems but also for applications in fields like astronomy and ballistics.