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Chapter 3: Problem 53

In fighting forest fires, airplanes work in support of ground crews by dropping water on the fires. A pilot is practicing by dropping 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 \(\mathrm{m}\) above the ground and with a speed of \(64.0 \mathrm{m} / \mathrm{s}(143 \mathrm{mi} / \mathrm{h}),\) at what horizontal distance from the target should the pilot release the canister? Ignore air resistance.

Short Answer

Expert verified
Release the canister 274.56 m from the target.

Step by step solution

01

Understand the Problem

The plane is flying horizontally at a height of 90 m and a speed of 64 m/s. We need to determine the horizontal distance from the target at which the pilot should release the canister so it lands on the target. Ignore air resistance.
02

Calculate Time of Fall

Find the time it takes for the canister to fall 90 meters. Use the formula for the time of fall: \[ t = \sqrt{\frac{2h}{g}} \]where \( h = 90 \) m and \( g = 9.8 \) m/s虏 (acceleration due to gravity). Substitute the values:\[ t = \sqrt{\frac{2 \times 90}{9.8}} \]
03

Solve for Time of Fall

Calculate the fall time:\[ t = \sqrt{\frac{180}{9.8}} \approx \sqrt{18.367} \approx 4.29 \text{ s}\]The canister will take approximately 4.29 seconds to hit the ground.
04

Calculate Horizontal Distance

Now that we know the time to fall, calculate the horizontal distance traveled by the canister during this time using the formula:\[ d = v \times t \]where \( v = 64 \) m/s and \( t = 4.29 \) s. Substitute the values:\[ d = 64 \times 4.29 \]
05

Solve for Horizontal Distance

Calculate the horizontal distance:\[ d = 64 \times 4.29 \approx 274.56 \text{ m} \]The pilot should release the canister approximately 274.56 meters from the target.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics
Kinematics is the branch of physics that deals with motion without considering the forces that cause it. Understanding kinematics allows us to predict the future position of an object based on its initial velocity, acceleration, and time of travel. It focuses on several key variables:
  • Displacement: The change in an object's position.
  • Velocity: The speed and direction of an object's motion.
  • Acceleration: The rate at which an object's velocity changes.
  • Time: The duration over which motion occurs.
In the exercise, the airplane is moving in a horizontal direction at a constant speed of 64 m/s. The canister is dropped from a height of 90 meters. Here, knowing how to manipulate the equations of motion can help determine the distance the canister travels. The key to solving problems like this is recognizing that horizontal and vertical motions can be treated independently of each other. Understanding kinematics is crucial to solving projectile motion problems effectively.
Horizontal Distance Calculation
The horizontal distance calculation is crucial in determining where to release the canister to ensure it hits the target. When the canister is released, it carries the velocity of the plane horizontally. Horizontally, the canister will continue its motion with the velocity it had at the release time, which is 64 m/s in this case.

Since the horizontal motion assumes no acceleration (ignoring air resistance), we use the formula:
\[ d = v imes t \]Where:
  • \(d\) is the horizontal distance
  • \(v\) is the velocity of the plane (64 m/s)
  • \(t\) is the time duration the canister is in the air (4.29 s)
The calculation already provided tells us that the horizontal distance is approximately 274.56 meters. This means the pilot should release the canister at this distance from the target to account for the time it takes to fall to the ground.
Free Fall
Free fall describes the motion of an object being acted upon only by the force of gravity. This occurs when objects fall under the influence of gravity alone, without any air resistance. In this scenario, the canister experiences free fall after being released from the plane.

To understand how long it takes to hit the ground, we use the formula for the time of fall:
\[ t = \sqrt{\frac{2h}{g}} \]Where:
  • \(h\) is the height from which the object is dropped (90 m)
  • \(g\) is the acceleration due to gravity (approximated as 9.8 m/s虏)
In this problem, the calculation of the time of fall reveals that the canister will take approximately 4.29 seconds to reach the ground from a 90-meter height. This time element is essential because it combines with the horizontal velocity to determine where to release the canister, aligning it to hit the target accurately.

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Most popular questions from this chapter

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