# A boat's acceleration is proportional to its velocity

• Skomatth
In summary, the boat's speed decreases when the engine is shut off due to the frictional force between the boat and water, which is proportional to the boat's speed. To find the time required for the boat to slow to 45 km/h, a differential equation for the boat's velocity can be used, where the velocity is proportional to itself with a negative constant. This can be solved using separation of variables or by finding a function that satisfies this equation. Additionally, the drag force on the boat is proportional to the boat's velocity squared, with different constants for water and air.
Skomatth
A 1000kg boat is traveling at 90km/h when its engine is shut off. The magnitude of the frictional force f between the boat and water is proportional to the speed v of the boat: f=70v where v is in meters per second and f is in Newtons. Find the time required for the boat to slow to 45 km/h.

This problem wasn't assigned so I might be trying something I'm not supposed to know how to do. I have a FBD and have defined the x-axis in the direction of the boat's motion.

-f=ma=-70v
therefore v(t)=$$-\int .070vdt$$

I wish I could show more work but I'm don't know where to go next. I think this employs some calculus I'm not familiar with so if someone could just point out a concept I need to look at I would appreciate it.

Write a=(dv/dt).
So, you have (dv/dt)=-kv, where k is a constant.

This is a "differential equation for v".
To solve for v, you have to find a function v that satisfies this equation. You don't need a class in differential equations to solve this, however.

Can you think of a function of t whose derivative is proportional [with a negative constant] to itself? If you can't you can try a technique called "separation of variables" to obtain such a function.

Thanks I got it.

To keep the problem real use drag force proportional to the velocity squared as:

Water Drag = 1/2 xCd x A x V^2 or proportional to V^2 with Cd and A constant.

Same functional relation for air drag except different Cd and A

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## What does it mean when it is said that a boat's acceleration is proportional to its velocity?

When it is said that a boat's acceleration is proportional to its velocity, it means that the boat's acceleration increases or decreases at a rate that is directly proportional to its velocity. This means that as the boat's velocity increases, its acceleration will also increase at the same rate.

## How does this relationship between acceleration and velocity affect the boat's motion?

The relationship between acceleration and velocity affects the boat's motion by determining how quickly the boat's speed will change over time. If the boat's acceleration is high, the boat will gain speed quickly. On the other hand, if the boat's acceleration is low, it will take longer for the boat to reach a higher speed.

## Is this relationship between acceleration and velocity always true for boats?

Yes, this relationship between acceleration and velocity is always true for boats. It follows the fundamental laws of physics, specifically Newton's Second Law, which states that an object's acceleration is directly proportional to the net force acting on it and inversely proportional to its mass.

## How does the mass of the boat affect its acceleration and velocity?

The mass of the boat affects its acceleration and velocity in the sense that a heavier boat will require more force to accelerate and will have a lower acceleration compared to a lighter boat. However, the relationship between acceleration and velocity remains the same, where acceleration is proportional to velocity.

## Can this relationship between acceleration and velocity be applied to other objects besides boats?

Yes, this relationship between acceleration and velocity can be applied to any object, as it follows the fundamental laws of physics. In fact, it can be seen in many real-life situations, such as a car's acceleration on a road or a person's acceleration while running. As long as the object is in motion, this relationship will hold true.

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