On a windless day, a sailor (m=75kg) decides he wants to go sailing (=300kg). In order to move on this windless day, he decides to mount a powerful battery operated fan )m=(20 kg) on the back of his sailboat and points the running fan at the sail. The fan can deliver air that pushes against the sail to the right with a constant force of 300N.
7A. If the sailboat is at rest and if the sailor runs the fan for 10 seconds, what will be the velocity of the boat at the end of 10 seconds?
7B. Would the sailor travel faster with with the sail Down or the sail up?
a. zero. Newtons third law.
b. down, as the fan will propel the ship, as in an airboat.
5.79
sail down
To calculate the velocity of the boat at the end of 10 seconds, we need to apply Newton's second law of motion.
7A. The formula we'll be using is F = m*a, where F is the applied force, m is the mass, and a is the acceleration.
Given:
Mass of the sailor (m1) = 75 kg
Mass of the boat (m2) = 300 kg
Mass of the fan (m3) = 20 kg
Force applied by the fan (F) = 300 N
Time (t) = 10 seconds
First, let's calculate the acceleration of the system. The fan is pushing the sail with a force of 300 N, and this force will result in an acceleration for the entire system.
Total mass (M) = m1 + m2 + m3 = 75 kg + 300 kg + 20 kg = 395 kg
Using the formula F = m*a, we can solve for a:
300 N = 395 kg * a
a = 300 N / 395 kg
a ≈ 0.76 m/s^2
Now, we can calculate the change in velocity (Δv) of the boat after 10 seconds using the equation v = u + a*t, where u is the initial velocity, t is the time, and a is the acceleration.
Initial velocity (u) = 0 m/s (since the boat is at rest)
v = u + a*t
v = 0 + 0.76 m/s^2 * 10 s
v = 7.6 m/s
Therefore, the velocity of the boat at the end of 10 seconds will be approximately 7.6 m/s.
7B. Intuitively, the sailor will travel faster with the sail up. When the sail is down, it will act as a barrier to the airflow and create resistance, reducing the net force exerted on the boat. With the sail up, the fan can push against it more effectively, resulting in a higher net force and therefore a higher velocity for the boat.
To answer these questions, we need to apply Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration.
In this case, the net force on the sailboat can be calculated as the force exerted by the fan minus the force of friction between the boat and the water. The force of friction can be assumed constant since the boat is on a windless day.
The acceleration of the boat can then be determined using the formula F = m * a, where F is the net force, m is the total mass of the boat and the fan, and a is the boat's acceleration.
For 7A, we can calculate the acceleration of the boat by dividing the net force (300N) by the combined mass of the boat and the fan (75kg + 300kg + 20kg). The result will give us the acceleration of the boat.
Once we have the acceleration, we can use the formula v = u + a * t to find the final velocity of the boat after the 10 seconds. Here, u is the initial velocity (which is zero since the boat is at rest), a is the acceleration, and t is the time duration, which is 10 seconds in this case.
For 7B, we need to consider the effect of the sail position on the velocity of the boat. When the sail is down, it will catch the air pushed by the fan more effectively, resulting in a larger force exerted on the boat. Therefore, the boat will accelerate faster with the sail down and ultimately reach a higher velocity compared to when the sail is up.
In conclusion:
7A. To find the velocity of the boat at the end of 10 seconds, calculate the net force by subtracting the force of friction from the force exerted by the fan. Then, use the formula v = u + a * t, where u is the initial velocity (which is zero), a is the acceleration, and t is the time duration (10 seconds).
7B. The sailor will travel faster with the sail down because it allows the boat to catch more air, resulting in a higher force and faster acceleration.