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A 1000kgboat is traveling at90km/h when its engine is shut off. The magnitude of the frictional forcefk between boat and water is proportional to the speed v of the boat:fk=70vwhere

vis in meters per second andfkis in Newton. Find the time required for the boat to slow to45km/h.

Short Answer

Expert verified

The time required for the boat to slow to45km/his9.9s

Step by step solution

01

Given

Mass of the boat, m=1000kg

Frictional force, fk=70vwherevis the speed of the boat.

Initial speed of the boat when engine is shut off,v0=90km/h

Final speed of the boat after time t, is

v=45km/h=v02km/h

02

Determining the concept

In this problem, the frictional force is not constant, but instead proportional to the speed of the boat. Integration is required to solve for the speed. Also, it involves Newton’s second law. According to Newton's 2nd law of motion, a force applied to an object at rest causes it to accelerate in the direction of the force.

Formula:

Fnet=ma

where, F is the net force, m is mass and a is an acceleration.

03

Determining the time required for the boat

Take the direction of the boat’s motion to be positive. Newton’s second law gives,

F=ma-fk=mdvdt-70v=mdvdtdvdt=-70mvdvv=-70mdt

Taking integration to both sides,

vovdvv=-70mt-0tdtInvv0=-70mtt=-m70Inv0/2v0=-100070-0.693=9.9

Hence, the time required for the boat to slow to45km/his9.9s

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