Problem 43 - Physics for Scientists and Engineers with Modern Physics

Question

The pilot of an airplane traveling \(170 \mathrm{~km} / \mathrm{h}\) wants to drop supplies to flood victims isolated on a patch of land \(150 \mathrm{~m}\) below. The supplies should be dropped how many seconds before the plane is directly overhead?

Step-by-Step Solution

Answer

The supplies should be released approximately 5.53 seconds before the plane is overhead.

Step 1: Understand the Problem

We need to determine the time it takes for supplies to fall 150 meters. We also know the airplane's speed is 170 km/h. Our goal is to calculate how many seconds before the plane is over the target, the supplies should be released.

Step 2: Convert Airplane Speed

The speed of the airplane needs to be converted from kilometers per hour to meters per second. Since 1 km = 1000 m and 1 hour = 3600 seconds, the conversion is \( \frac{170 \, \text{km/h} \times 1000 \, \text{m/km}}{3600 \, \text{s/h}} = \frac{170000}{3600} \, \text{m/s} \approx 47.22 \, \text{m/s} \).

Step 3: Use Free Fall Formula

The formula for the time \( t \) it takes for an object to fall a certain distance \( h \) under gravity is \( t = \sqrt{\frac{2h}{g}} \), where \( g \approx 9.8 \, \text{m/s}^2 \) is the acceleration due to gravity. Substitute \( h = 150 \, \text{m} \) into the formula to find the time the supplies take to fall.\[ t = \sqrt{\frac{2 \times 150}{9.8}} \approx \sqrt{30.61} \approx 5.53 \, \text{s} \]

Step 4: Calculate Drop Time Before Overhead

Since it takes roughly 5.53 seconds for the supplies to fall, they should be released approximately 5.53 seconds before the plane is directly overhead.

Key Concepts

Free Fall

Free fall is a simple concept in physics where an object that is dropped is only acted upon by gravitational force. This means air resistance or other forces aren't considered in the calculation of an object under free fall.
When an object is in free fall, it experiences a constant acceleration due to gravity, which on Earth is approximately 9.8 m/s².
As a result, an object in free fall will continue to increase its speed as it descends.

• Free fall does not depend on the mass of the object. All objects dropped from the same height will hit the ground at the same time, assuming no air resistance.
• The only force acting on an object in free fall is gravity, making it a crucial assumption for many physics problems like dropping supplies from an airplane.

Kinematics Equations

Kinematics equations are used to describe the motion of objects under the influence of constant acceleration.
They help us calculate positions, velocities, and times, given known initial values and accelerations.
In the exercise about dropping supplies from an airplane, the kinematics equation \[ t = \sqrt{\frac{2h}{g}} \]is used, where:

• \(t\) is the time in seconds
• \(h\) is the height the object falls
• \(g\) is the acceleration due to gravity

This equation is derived from the basic kinematics formula for constant acceleration, indicating how the time for falling relates to the height and gravitational acceleration.
Understanding these equations allows you to connect velocity, position, and acceleration in a clear, mathematical way.
They are incredibly useful for solving real-world problems involving any object in motion, including things like launching rockets or the flight of cars off ramps.

Acceleration due to Gravity

Acceleration due to gravity, symbolized by \(g\), is the rate at which objects are pulled towards the center of the Earth.
On Earth, \(g\) is approximately 9.8 m/s².
This value is crucial when calculating the time it takes for objects to fall in free fall.

• In every second, the velocity of an object in free fall increases by 9.8 meters per second.
• Gravity acts uniformly on all objects, providing the constant acceleration needed for kinematics.

The constant \(g\) allows us to make accurate predictions about how long it will take for an object to hit the ground in absence of any other forces.
It's important to note that while \(g\) is often rounded to 9.8 m/s² for simplicity, slight variations exist depending on altitude and geographical location on Earth.