) An oil tanker’s engines have broken down, and the wind is blowing the tanker straight toward a reef at a constant
speed of 1.5 m/s (Fig.1). When the tanker is 500 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 * 107 kg, and the engines produce a net horizontal force of 8.0 * 104 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 m/s or less. Ignore the retarding force of the water on the tanker’s hull.
To determine whether the oil tanker will hit the reef, we need to calculate the time it takes for the tanker to reach the reef at a constant speed of 1.5 m/s.
Step 1: Find the time it takes to reach the reef.
Using the formula:
Time = Distance / Speed
Given:
Distance = 500 m
Speed = 1.5 m/s
Time = 500 m / 1.5 m/s
Time = 333.33 seconds
Step 2: Calculate the acceleration required to stop the ship within the given distance of 500 m.
Using the formula:
Acceleration = Final Velocity^2 / (2 * Distance)
Given:
Final Velocity = 0 m/s
Distance = 500 m
Acceleration = (0 m/s)^2 / (2 * 500 m)
Acceleration = 0 m/s^2
Step 3: Calculate the net force required to produce the required acceleration.
Using Newton's second law:
Force = Mass * Acceleration
Given:
Mass of the tanker and cargo = 3.6 * 10^7 kg
Acceleration = 0 m/s^2
Force = (3.6 * 10^7 kg) * (0 m/s^2)
Force = 0 N
Comparing the net force produced by the engines (8.0 * 10^4 N) with the force required to stop the tanker within the given distance (0 N), we can see that the force produced is greater than the force required. Hence, the ship will not hit the reef.
However, it's worth noting that this analysis doesn't account for the rudder being stuck, which may affect the ship's ability to accelerate backward. Additionally, we haven't addressed the safety of the oil in the event of an impact since we don't have enough information about the oil or the impact forces.
To determine if the ship will hit the reef and if the oil will be safe, we need to analyze the forces acting on the ship.
Initially, the ship is being pushed towards the reef by the wind with a constant speed of 1.5 m/s. This means that there is no net force acting horizontally to oppose the motion of the ship. However, the engines produce a net horizontal force of 8.0 * 10^4 N on the ship.
To find the acceleration of the ship, we can use Newton's second law of motion, which states that the net force is equal to the mass of an object multiplied by its acceleration. Rearranging the equation, we have:
Acceleration (a) = Net force (F) / Mass (m)
Substituting the given values, we have:
a = (8.0 * 10^4 N) / (3.6 * 10^7 kg) = 2.22 * 10^-3 m/s^2
Now, we can determine the time it takes for the ship to reach the reef from its initial position of 500 m. We can use the following equation of motion, assuming constant acceleration:
s = ut + (1/2)at^2
Where:
s = distance traveled (500 m)
u = initial velocity (1.5 m/s)
a = acceleration (-2.22 * 10^-3 m/s^2, negative because it opposes the motion)
t = time
Rearranging the equation, we have:
(1/2)at^2 + ut - s = 0
Solving this quadratic equation, we get two possible values for t: t = 148.88 s or t = -148.88 s. Since time cannot be negative, we discard the negative value.
Hence, it will take approximately 148.88 seconds for the ship to reach the reef.
Next, we need to determine the final velocity of the ship when it reaches the reef. We can use another equation of motion:
v = u + at
Substituting the given values:
v = (1.5 m/s) + (-2.22 * 10^-3 m/s^2) * (148.88 s) = 1.16 m/s
The final velocity of the ship when it reaches the reef is approximately 1.16 m/s.
Since the ship's speed is less than the maximum impact speed the hull can withstand (0.2 m/s), the ship will not cause any damage to the reef or be at risk of hull damage.
Therefore, the ship will not hit the reef, and the oil will remain safe.
Well, whatever we do we will be wrong because when you have a ship in water there is an effect called usually "added mass" due to having to accelerate water around the ship when you accelerate the ship. However your book did not know that so:
F = m a
-8*10^4 = 3.6*10^7 a
so
a = - (8/3.6)10^-3 m/s^2
v = 1.5 m/s + a t
when v = 0
t = (1.5*3.6*10^3)/8 s
that is the time to stop
d = 1.5 t -(1/2)(8/3.6)10^-3 t^2
that is the distance to stop