The force exerted by the wind on a sailboat is approximately perpendicular to the sail and proportional to the component of the wind velocity perpendicular to the sail. For the 792-kg sailboat shown in the figure below, the force exerted by the wind on the sailboat is
Fsail =
550
N
m/s
vwindorthogonal.
Water exerts a force along the keel (bottom) of the boat that prevents it from moving sideways, as shown in the figure. Once the boat starts moving forward, water also exerts a drag force backwards on the boat, opposing the forward motion. If a 16-knot wind (1 knot = 0.514 m/s) is blowing to the east, what is the initial acceleration of the sailboat?
m/s2
To find the initial acceleration of the sailboat, we need to consider the forces acting on it.
First, let's break down the given information:
- The force exerted by the wind on the sailboat is given by Fsail = (550 N/m/s) * vwindorthogonal.
- We are given that the wind velocity is 16 knots, which we can convert to meters per second by multiplying by 0.514: vwind = 16 knots * 0.514 m/s/knot = 8.192 m/s.
Now, let's analyze the forces acting on the sailboat:
1. Force exerted by the wind (Fsail): This force is perpendicular to the sail and proportional to the component of the wind velocity perpendicular to the sail. This force will provide the forward acceleration of the sailboat.
2. Force exerted by the water: The water exerts a force along the keel to prevent sideways movement and also exerts a drag force opposing the forward motion. Since we are asked for the initial acceleration, we can ignore the drag force at this point.
Since the only external force acting on the sailboat is the force exerted by the wind, we can equate it to the product of the mass of the sailboat (792 kg) and its acceleration (a):
Fsail = mass * acceleration
Substituting the values we know:
550 N/m/s * vwindorthogonal = 792 kg * a
To find the acceleration, we need to determine the wind component perpendicular to the sail (vwindorthogonal). Since the wind is blowing to the east and the sailboat is moving forward, the component of the wind perpendicular to the sail is also the component of the wind in the eastward direction.
Thus, the wind velocity perpendicular to the sail (vwindorthogonal) is equal to the wind velocity (vwind) of 8.192 m/s.
Now, plug in the values:
550 N/m/s * 8.192 m/s = 792 kg * a
Solving for acceleration (a):
a = (550 N/m/s * 8.192 m/s) / 792 kg
a ≈ 5.69 m/s^2
Therefore, the initial acceleration of the sailboat is approximately 5.69 m/s^2.