On a windless day, a sailor (m = 75 kg) decides he wants to go sailing in his sailboat (m = 300 kg). 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 ponits the running fan at the sail. The fan can deliver air that pusher against the sail to the right with a constant force of 300 N.
1. If the sailboat is at rest and if the sailor runs the fan for 10 s, what will be the velocity of the boat at the end of the 10 seconds?
2. Would the sailor travel faster with the sail fown or the sail up?
1. To find the velocity of the boat at the end of 10 seconds, we can use Newton's second law of motion, which relates the force applied to an object, its mass, and its acceleration. The formula for force is:
Force = mass x acceleration
The mass of the boat and sailor combined is 375 kg (300 kg + 75 kg). The mass of the fan is irrelevant for this calculation. The force applied by the fan is 300 N, and we can assume there is no friction or other external forces affecting the boat's motion.
Since we know the force and mass, we can rearrange the formula to solve for acceleration:
Acceleration = Force / mass
Plugging in the values:
Acceleration = 300 N / 375 kg
Acceleration = 0.8 m/s^2
Now, we can use the equation of motion:
Final velocity = Initial velocity + (Acceleration x Time)
Since the boat is initially at rest, the initial velocity is 0 m/s. Plugging in the values:
Final velocity = 0 + (0.8 m/s^2 x 10 s)
Final velocity = 8 m/s
Therefore, the velocity of the boat at the end of the 10 seconds is 8 m/s.
2. The sailor would travel faster with the sail up rather than with the sail down. When the sail is up, it can catch the wind and convert its energy into forward motion. This is known as harnessing the power of the wind. On the other hand, when the sail is down, the boat's only source of propulsion is the fan. Although the fan can generate a force, it is limited compared to the force generated by the wind. Therefore, the sail up position allows the boat to take advantage of the available wind and move faster.
To answer these questions, we can apply Newton's second law, which states that the net force acting on an object is equal to the product of its mass and acceleration (F = ma). In this case, we need to find the acceleration of the sailboat in order to determine its velocity.
1. To calculate the velocity of the boat at the end of 10 seconds, we can start by finding the acceleration it experiences due to the force applied by the fan. Given that the mass of the boat is 300 kg and the force exerted by the fan is 300 N, we can use the formula F = ma to find the acceleration:
F = ma
300 N = (300 kg) * a
Solving for acceleration:
a = 300 N / 300 kg = 1 m/s^2
Now, we can use the kinematic equation:
v = u + at
where:
v = final velocity (what we want to find)
u = initial velocity (0 m/s in this case since the boat is at rest)
a = acceleration (1 m/s^2)
t = time (10 s)
Plugging in the values, we get:
v = 0 + (1 m/s^2) * (10 s)
v = 10 m/s
Therefore, the velocity of the boat at the end of the 10 seconds would be 10 m/s.
2. Whether the sailor would travel faster with the sail up or down depends on the presence of wind. On a windless day, the sail would not have any effect on the boat's speed. The force provided by the battery-operated fan is what is propelling the boat forward. Hence, in this scenario, the sail position (up or down) would not affect the boat's speed.