Problem 38 - University Physics with Modern Physics

Question

An oil tanker's engines have broken down, and the wind is blowing the tanker straight toward a recf at a constant spocd of 1.5 \(\mathrm{m} / \mathrm{s}(\text { Fig. } 4.37) .\) When the tanker is 500 \(\mathrm{m}\) m from the reef, the wind dies down just as the engineer gets the engines going again. The rudder is stuck, so the only choice is to try to accelerate straight backward away from the reef. The mass of the tanker and cargo is \(3.6 \times 10^{7} \mathrm{kg}\) , and the engines produce a net horizontal force of \(8.0 \times 10^{4} \mathrm{N}\) on the tanker. Will the ship hit the reef? If it does, will the oil be safe? The hull can withstand an impact at a speed of 0.2 \(\mathrm{m} / \mathrm{s}\) or less. You can ignore the retarding force of the water on the tanker's hull.

Step-by-Step Solution

Answer

The ship will stop before hitting the reef; the oil will be safe.

Step 1: Define the Known Values

First, identify the given quantities in the problem:- Initial speed of the tanker, \( v_i = 1.5 \, \mathrm{m/s} \).- Distance to the reef, \( d = 500 \, \mathrm{m} \).- Force from the engines, \( F = 8.0 \times 10^4 \, \mathrm{N} \).- Mass of the tanker, \( m = 3.6 \times 10^7 \, \mathrm{kg} \).- Maximum safe impact speed, \( v_{ ext{safe}} = 0.2 \, \mathrm{m/s} \).

Step 2: Calculate the Acceleration

We'll calculate the acceleration produced by the force using Newton's second law:\[ a = \frac{F}{m} = \frac{8.0 \times 10^4 \, \mathrm{N}}{3.6 \times 10^7 \, \mathrm{kg}} = 2.22 \times 10^{-3} \, \mathrm{m/s^2} \]

Step 3: Determine the Time Needed to Stop

Use the formula for time given initial velocity and constant acceleration:\[ t = \frac{v_i}{a} = \frac{1.5 \, \mathrm{m/s}}{2.22 \times 10^{-3} \, \mathrm{m/s^2}} \approx 675.68 \, \mathrm{s} \]

Step 4: Determine the Stopping Distance

Use the kinematic equation to calculate the stopping distance:\[ d_{ ext{stop}} = v_i \cdot t + \frac{1}{2} a t^2 \]Substitute \( t = 675.68 \, \mathrm{s} \) into the equation:\[ d_{\text{stop}} = 1.5 \cdot 675.68 + \frac{1}{2} \times 2.22 \times 10^{-3} \times (675.68)^2 \approx 1013.52 \, \mathrm{m} \]

Step 5: Compare Stopping Distance with 500 m

Since the stopping distance \( d_{\text{stop}} = 1013.52 \, \mathrm{m} \) is greater than 500 m, the tanker will stop before reaching the reef.

Step 6: Calculate the Impact Speed (if it were needed)

Since the stopping distance is more than 500 m, the tanker will not hit the reef. If it were to hit, it would eventually stop by the time it still has a safe speed left for calculation.

Key Concepts

Kinematics

In physics, kinematics is the study of motion, without focusing on the forces that cause it. In our example with the oil tanker, we begin by determining its initial speed as it drifts towards the reef. The tanker moves at a constant speed of 1.5 m/s in the absence of any opposing forces initially.
This speed is crucial for calculating how far and how fast over time the tanker will move. Due to the rudder being stuck and the immediate stopping of wind, our problem reduces to a kinematics problem where we compute the stopping distance with a source of backward acceleration.
The basic set of equations called the kinematic equations help us to relate the initial velocity, final velocity, distance, acceleration, and time. Here, knowing the initial velocity, distance, and the required stopping speed directs the solution by leveraging these equations. We use them to derive the time required to stop and the distance covered in that period.

Newton's Laws of Motion

Newton's Laws of Motion form the bedrock of classical mechanics, providing insights into how objects behave under various forces. For the oil tanker problem, Newton's Second Law (F = ma) plays a crucial role. This law states that force equals mass times acceleration and is pivotal when analyzing the impact of applied forces on an object.
In our scenario, the engine exerts a force of \(8.0 \times 10^4 \, \mathrm{N}\) to counter the tanker's motion towards the reef. Using Newton's Second Law, we compute the resulting acceleration. Despite the tanker's massive weight (\(3.6 \times 10^7 \, \mathrm{kg}\)), this force allows the tanker to decelerate, thus heading backward from the reef.
Understanding Newton's Second Law helps in calculating how quickly the tanker can stop when a force is applied in the opposite direction to its motion. Recognizing how forces and mass interrelate underscores essential concepts you'll frequently encounter in physics.

Acceleration and Force

Acceleration is the rate of change of velocity per unit time, directly influenced by the force applied to an object according to Newton's Second Law. Regardless of the tanker's large mass, the backward force applied by the engines is enough to yield an acceleration of \(2.22 \times 10^{-3} \, \mathrm{m/s^2}\).
In practical scenarios like this, understanding the relationship between force and acceleration allows you to determine how effectively an engine or any forceful action can influence a large body. The computed acceleration tells us how quickly the tanker will change velocity, either speeding up or slowing down.
Here, acceleration helps us determine how long it will take to bring the tanker to a complete stop, thus averting a collision. Deciphering the magnitude of this acceleration alongside the limitations of force accelerates understanding of key concepts in physics and helps in predicting future behavior.

Distance and Time Calculations

Effective problem-solving in physics often hinges on the ability to calculate distances and times accurately. In our example, the tanker starts 500 meters away from the reef. We must calculate both the time and distance needed for it to come to a halt.
Using kinematic equations, we've already established that the tanker requires about 675.68 seconds to stop, and during this interval, it covers a distance of approximately 1013.52 meters. These calculations highlight that the tanker will stop well before reaching the reef.
Such calculations also provide an insight on if the tanker were closer, how close it can get before the speed reduction becomes essential. This forms the crux of ensuring safety in navigation scenarios and illustrates the practical applications of physics in real-world problems.