Problem 47 - University Physics with Modern Physics

Question

In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. For practice, a pilot drops a canister of red dye, hoping to hit a target on the ground below. If the plane is flying in a horizontal path 90.0 m above the ground and has a speed of 64.0 m/s (143 mi/h), at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

Step-by-Step Solution

Answer

The canister should be released approximately 274.56 m from the target.

Step 1: Identify Given Values

We are given the following values:- Height from which the canister is dropped: \( h = 90.0 \; \text{m} \)- Horizontal speed of the plane: \( v = 64.0 \; \text{m/s} \)We are asked to find the horizontal distance \( d \) from the target at which the canister should be released.

Step 2: Determine Time of Fall

First, we need to find the time it takes for the canister to hit the ground. This can be determined using the equation for free fall: \[ h = \frac{1}{2} g t^2 \]Where:- \( h = 90.0 \; \text{m} \)- \( g = 9.8 \; \text{m/s}^2 \) (acceleration due to gravity)Rearranging the equation to solve for \( t \):\[ t^2 = \frac{2h}{g} \]\[ t = \sqrt{\frac{2h}{g}} \]

Step 3: Calculate the Time of Fall

Substitute the known values into the time of fall equation:\[ t = \sqrt{\frac{2 \times 90.0}{9.8}} \]\[ t = \sqrt{\frac{180.0}{9.8}} \]\[ t = \sqrt{18.367} \]\[ t \approx 4.29 \; \text{s} \]

Step 4: Calculate Horizontal Distance

Now, use the horizontal speed of the plane and the time calculated to find the horizontal distance:\[ d = v \times t \]Where:- \( v = 64.0 \; \text{m/s} \)- \( t \approx 4.29 \; \text{s} \)Substitute these values into the equation:\[ d = 64.0 \times 4.29 \]\[ d \approx 274.56 \; \text{m} \]

Step 5: Conclusion and Checking

The horizontal distance from the target is indeed calculated to be approximately \( 274.56 \; \text{m} \). Check if it was computed based on the correct time of fall and proper substitution. This confirms that the canister should be released when the plane is about \( 274.56 \; \text{m} \) away horizontally from the target.

Key Concepts

Free Fall

Free fall is a type of motion where an object is subjected to only the force of gravity. This means we do not take into account any other forces, like air resistance. In the scenario where the canister is dropped from an airplane, the only force acting on it is due to gravity. When you hear the term "free fall," it's essential to remember that objects in free fall experience constant acceleration due to the gravitational force. Because gravity is a uniform force, the object accelerates at a steady rate of approximately 9.8 m/s² downwards. This happens regardless of the object’s initial state of motion.To calculate the time it takes for the canister to hit the ground, we use the formula:\[ h = \frac{1}{2} g t^2 \]where:- \( h \) is the height from which the object is dropped.- \( g \) is the acceleration due to gravity.- \( t \) is the time taken to fall.By rearranging this equation, we can solve for \( t \), which helps us determine how long the canister is in free fall. Understanding free fall is crucial because it allows us to calculate other factors like horizontal distance, knowing how much time elapses while the object is in motion.In summary, free fall helps us calculate the time of descent when only gravity is at play, serving as a vital building block for solving projectile motion problems.

Horizontal Distance Calculation

When an object like a canister is dropped from a moving airplane, it travels both horizontally and vertically. While the vertical movement is affected by gravity, the horizontal movement remains constant since we ignore air resistance here.To find out how far the canister travels horizontally before hitting the ground, we calculate the horizontal distance using the simple formula:\[ d = v \times t \]Here:- \( d \) is the horizontal distance.- \( v \) is the horizontal speed of the airplane.- \( t \) is the time the canister is in free fall, which we found previously using the concept of free fall.The critical point to remember is that horizontal speed is constant throughout the motion, as there are no forces acting horizontally. This means any point during its travel should have the same speed horizontally.By multiplying the horizontal speed by the time of fall, we determine how far the canister will have traveled on the horizontal plane by the time it reaches the ground. This calculation is crucial for determining the point at which to release the canister to hit the target accurately.

Acceleration due to Gravity

The concept of acceleration due to gravity is a central component of projectile motion calculations and is universally denoted as \( g \). In the context of solving the problem at hand, the value of \( g \) is 9.8 m/s², which is considered as a constant for all objects in free fall near the Earth's surface.Gravity is responsible for accelerating the canister downward as soon as it is released from the airplane. Each second, the object's downward velocity increases by approximately 9.8 meters per second, which directly influences how fast it hits the ground and, consequently, how long it is in free fall.This natural phenomenon plays a vital role in both the vertical motion and the prediction of where the canister will land. Without considering gravity, calculations of fall time and corresponding horizontal distances would be impossible.Remember that gravity acts uniformly on all objects, regardless of their mass, as long as air resistance is negligible. This universality makes gravity a reliable factor in physics calculations, allowing precise and predictable outcomes when dealing with free fall and projectile motion.