A 1000 kg boat is traveling at 100 km/h when its engine is...

Name: Problem 33, Chapter 6: Principles of Physics
Uploaded: 2022-03-19T12:19:10-08:00
Duration: 3 min 39 s
Channel: Sandeep Kumar Dhania
Description: A 1000 kg boat is traveling at 100 km/h when its engine is shut off. The magnitude of the frictional force between boat and water is proportional to the speed v of the boat:fk 70v, where v is in meters per second and fk is in newtons. Find the time required for the boat to slow to 45 km/h.

A 1000 kg boat is traveling at 100 km/h when its engine is shut off. The magnitude of the frictional force between boat and water is proportional to the speed v of the boat:fk 70v, where v is in meters per second and fk is in newtons. Find the time required for the boat to slow to 45 km/h.

A 1000 kg boat is traveling at 100 km/h when its engine is shut
off. The magnitude of the frictional force between boat and water is proportional to the speed v of the boat:fk 70v, where v is in
meters per second and fk is in newtons. Find the time required for
the boat to slow to 45 km/h.

Key Concepts

Newton's Second Law

Newton's Second Law states that the acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass. This principle is used to relate the frictional force acting on the boat to its deceleration, allowing the formulation of a differential equation that models the boat’s motion as it slows down.

Resistive Force Proportional to Speed

When a resistive or frictional force is proportional to the speed of an object, it implies that the deceleration experienced by the object increases linearly with its velocity. This type of frictional force leads to an exponential decay in speed, which is described mathematically using a first-order linear differential equation.

Differential Equations and Separation of Variables

The motion of the boat under a speed-proportional frictional force is modeled using a differential equation where the rate of change of velocity is proportional to the velocity itself. This differential equation can be solved using the method of separation of variables, which involves rearranging the equation so that all terms involving velocity are on one side and all terms involving time are on the other, and then integrating both sides.

Exponential Decay

Exponential decay describes processes where the rate of change of a quantity is proportional to the current value of that quantity. In the context of this problem, the boat’s velocity decreases exponentially over time under the influence of a resistive force that is proportional to its speed, leading to a decay function that can be used to determine the time required for the boat to slow from one specified speed to another.

Unit Conversion

Accurate unit conversion is essential in solving physics problems, especially when different units are used for various quantities. In this problem, speeds are initially given in kilometers per hour and need to be converted to meters per second in order to be consistent with the units used in the frictional force expression and the resulting differential equation.

Transcript

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