Problem 17 - Thomas Calculus

Question

Sailing A sailboat is running along a straight course with the wind providing a constant forward force of 50 lb. The only other force acting on the boat is resistance as the boat moves through the water. The resisting force is numerically equal to five times the boat's speed, and the initial velocity is 1 \(\mathrm{ft} /\) sec. What is the maximum velocity in feet per second of the boat under this wind?

Step-by-Step Solution

Answer

The maximum velocity of the boat is 10 ft/s.

Step 1: Identify Forces Involved

First, identify the forces acting on the sailboat. The boat is subjected to a forward force due to wind, which is constant at 50 lb. Additionally, there is a resisting force due to the water, which is given by the formula 5v, where v is the speed of the boat in feet per second.

Step 2: Set Up Force Balance Equation

To find the condition for maximum velocity, set the net force on the boat to zero (as it reaches constant maximum speed). Write the equation: Forward force - resisting force = net force, which gives us: 50 - 5v = 0.

Step 3: Solve for Maximum Velocity

Solve the equation 50 - 5v = 0 for v. Rearrange the equation to find the velocity: 5v = 50, which simplifies to v = 50/5.

Step 4: Calculate the Maximum Velocity

Divide 50 by 5 to determine the maximum velocity: v = 10 ft/s.

Key Concepts

Force Balance Equation

A force balance equation is a fundamental concept in physics used to understand equilibrium states. It involves summing all the forces acting on an object and setting them equal to identify conditions, such as maximum velocity or rest. In our sailboat example, two main forces act on the boat: the wind providing a constant forward push of 50 lb and a resisting force from the water that opposes the motion.
To find when the boat reaches its maximum speed, we apply the force balance equation. By setting the net force to zero, because no further acceleration occurs at maximum speed, we write the equation as:

• Forward force (wind force) = 50 lb
• Resisting force (water) = 5v (where \(v\) is velocity in ft/s)
• Net force = Forward force - Resisting force = 0

This leads us to the equation: \[50 - 5v = 0\]This equation helps determine the boat's speed where the forces balance perfectly.

Maximum Velocity

Maximum velocity in this context is the constant speed that the sailboat can achieve under the given force conditions. When the forces acting on the boat balance out, it stops accelerating and moves at a consistent speed. This occurs when the net force is zero.
Considering our earlier force balance equation:\[50 - 5v = 0\]we solve for \(v\) (the speed of the boat) to find:

• Rearrange the equation to 5v = 50
• Divide by 5 to isolate v: \(v = 10\) ft/s

Therefore, the maximum velocity of the sailboat, under a constant wind force of 50 lb, is 10 feet per second.

Net Force

Net force is the total force acting on an object when all individual forces are combined. It dictates whether an object will accelerate, decelerate, or move at a constant velocity. In our scenario with the sailboat, the net force indicates changes in velocity.
When finding maximum velocity, the net force becomes zero, meaning the forces that push forward and backward are equal. The forward force from the wind is 50 lb, while the resisting force from the water is 5\(v\). At net force zero:

• Calculate net force: 50 - 5v = 0
• Indicates no acceleration when net force = 0

This balanced net force results in a steady velocity, crucial for understanding dynamics in physics.

Resistance Force

Resistance force is the opposing force that an object encounters when moving through a medium. For the sailboat moving across water, this force is comparable to friction, which slows down or opposes motion. It is proportionate to the speed of the object, here the boat, hence, described as a linear function of speed.
In our sailboat problem, the resistance force is given as 5\(v\), where \(v\) is the velocity of the boat in feet per second. This means every increase in speed results in greater opposition:

• Resisting force formula: 5\(v\)
• Greater the speed \(v\), greater the resisting force

This friction-like force must be overcome by the propelling force to accelerate. At maximum speed, it exactly equals the forward pushing force, balancing the forces and ensuring maximum velocity is sustained.