Problem 16 - College Physics

Question

\(\bullet(\) a) \(A\) pistol that fires a signal flare gives it an initial velocity (muzzle velocity) of 125 \(\mathrm{m} / \mathrm{s}\) at an angle of \(55.0^{\circ}\) above the horizontal. You can ignore air resistance. Find the flare's maximum height and the distance from its firing point to its landing point if it is fired (a) on the level salt flats of Utah, and (b) over the flat Sea of Tranquility on the moon, where \(g=1.67 \mathrm{m} / \mathrm{s}^{2}\) .

Step-by-Step Solution

Answer

(a) Maximum Height on Earth: 533.55 m, Range: 1496.11 m. (b) Maximum Height on Moon: 3136.23 m, Range: 8791.17 m.

Step 1: Determine Components of Initial Velocity

We need to break down the initial velocity of 125 m/s into its horizontal and vertical components. The horizontal component \( v_{x} \) can be found using \( v_{x} = v_0 \cos \theta \) and the vertical component \( v_{y} \) using \( v_{y} = v_0 \sin \theta \). For \( \theta = 55.0^{\circ} \), we have:\[ v_{x} = 125 \cos 55.0^{\circ} \approx 71.71 \text{ m/s} \]\[ v_{y} = 125 \sin 55.0^{\circ} \approx 102.29 \text{ m/s} \]

Step 2: Calculate Maximum Height (Earth)

The maximum height is reached when the vertical velocity component becomes zero. Use the equation \( v_{y}^2 = v_{0y}^2 - 2gh \), setting \( v_y = 0 \). Rearrange to find \( h \):\[ 0 = v_{y}^2 - 2gh \Rightarrow h = \frac{v_{y}^2}{2g} \]Substitute \( g = 9.81 \text{ m/s}^2 \):\[ h = \frac{(102.29)^2}{2 \times 9.81} \approx 533.55 \text{ m} \]

Step 3: Calculate Time of Flight (Earth)

The total time of flight can be calculated using the equation for the vertical motion: \( t = \frac{2v_{y}}{g} \). Substituting for Earth:\[ t = \frac{2 \times 102.29}{9.81} \approx 20.87 \text{ s} \]

Step 4: Calculate Range (Earth)

The range of the projectile is found using the horizontal speed and total flight time: \( R = v_{x} \times t \).\[ R = 71.71 \times 20.87 \approx 1496.11 \text{ m} \]

Step 5: Calculate Maximum Height (Moon)

Follow the same process for maximum height on the Moon where \( g = 1.67 \text{ m/s}^2 \):\[ h = \frac{(102.29)^2}{2 \times 1.67} \approx 3136.23 \text{ m} \]

Step 6: Calculate Time of Flight (Moon)

Calculate the flight time using the reduced gravity on the Moon:\[ t = \frac{2 \times 102.29}{1.67} \approx 122.62 \text{ s} \]

Step 7: Calculate Range (Moon)

Finally, find the range on the Moon:\[ R = 71.71 \times 122.62 \approx 8791.17 \text{ m} \]

Key Concepts

Kinematics

Kinematics is a foundational concept in physics that deals with the motion of objects without considering the forces causing the motion. When studying projectile motion, like the signal flare in the given exercise, understanding kinematics is crucial.

Projectiles are objects thrown into the air that follow a curved path under the influence of gravity. Their motion can be split into horizontal and vertical components. These components are addressed separately using kinematic equations.

Important elements in kinematic analysis include initial velocity, angle of launch, time of flight, maximum height, and range. Breaking down the initial velocity into horizontal and vertical components allows us to solve many projectile motion problems simply.

• Horizontal motion is uniform, with constant velocity since air resistance is ignored.
• Vertical motion is uniformly accelerated, due to gravity, affecting the speed and height of the projectile.

Understanding these concepts allows one to predict where and when a projectile will land.

Maximum Height

The maximum height of a projectile is the highest point it reaches during its flight. This occurs when the vertical component of its velocity (the speed at which it's moving upwards or downwards) becomes zero.

To find the maximum height, you can use the equation:
\
\[ h = \frac{v_{y}^2}{2g} \]
where \( v_{y} \) is the initial vertical velocity and \( g \) is the acceleration due to gravity.

• On Earth, gravity is approximately \( 9.81 \text{ m/s}^2 \).
• On the Moon, gravity is much weaker, at around \( 1.67 \text{ m/s}^2 \). This means projectiles go much higher and stay in the air longer on the Moon.

In our example, the maximum height over Utah was about 534 meters, but the flare would reach over 3136 meters on the Moon due to the lower gravitational pull.

Range of Projectile

The range of a projectile is the horizontal distance it travels from its launch point to where it lands. It's calculated by considering both the horizontal velocity and the total time it remains in the air.

The formula used is:
\
\[ R = v_{x} \times t \]
where \( v_{x} \) is the constant horizontal velocity and \( t \) is the total time of flight.

In this problem, by using the kinematic equations and considering the effects of gravity, we discover:

• The signal flare's range in Utah is about \( 1496 \text{ m} \).
• On the Moon, with weaker gravity, the flare travels much farther, covering approximately \( 8791 \text{ m} \).

Understanding the range helps predict the landing zone of projectiles, which is valuable for various applications like targeting in ballistics or sports.

Gravity Effects

Gravity plays a significant role in determining the path of any projectile. It is the force that pulls objects toward the center of celestial bodies like Earth and the Moon.

Due to gravity:

• The vertical motion of projectiles is accelerated, causing them to eventually stop ascending and begin descending.
• The time a projectile spends in the air and its maximum height depend directly on the strength of the gravitational pull.

For instance, on Earth, where gravity is stronger, projectiles don't reach as high or travel as far, with a flight time of about 21 seconds for our example.

On the Moon, where gravity is much weaker (about one-sixth of Earth's gravity), the same projectile reaches significantly higher and travels further, with a flight time of around 123 seconds. This explains why astronauts can jump higher and why missiles or other projectiles need different considerations when planning flights on the Moon.

Physics Problem Solving

Solving physics problems, particularly those involving projectile motion, involves logical steps and an understanding of key principles. This approach helps simplify complex problems into manageable parts:

• Break Down the Motion: Separate into horizontal and vertical components. This simplifies calculations.
• Use Kinematic Equations: These equations relate velocity, time, and distance, letting us find unknown values.
• Consider Gravity: Gravity affects the vertical motion, so always consider the gravitational force in calculations.
• Calculate Individual Components: Work out one part of the motion at a time to avoid confusion.

In the exercise, these steps allowed us to find the flare's maximum height and range on both Earth and the Moon. By practicing these techniques, you can tackle similar physics problems effectively, reinforcing your understanding of motion and forces.