Question

The speed of a motorboat in still water is four times the speed of a river. Normally, the motor boat takes one minute to cross the river to the port straight across on the other bank. One time, due to a motor problem, it was not able to run at full power, and it took four minutes to cross the river along the same path. By what factor was the speed of the boat in still water reduced? (Assume that the speed of the water is uniform throughout the whole width of the river.)

Answer 1

Explanation:

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Answer 2

The speed of the motorboat gets reduced by a factor of 16.

Given,

Time taken to cross the river when both the speed of river and that of the motorboat are operational=1 minute

Time taken to cross the river when only the speed of the river is operational and the speed of the motorboat got reduced=4 minutes

Speed of motorboat=4 x speed of the river.

To find,

the factor by which the speed of the boat in still water reduced.

Solution:

• The motion between the river and the motorboat is a downstream motion.
• In a downstream motion, the resultant speed of the motion is equal to the sum of the individual speeds of the components present in the motion.
• Vnet=V1+V2.

First case: When both the speed of river and that of the motorboat are operational.

Let the speed of the river,Vr be V.

Speed of the motorboat, Vb=4V.

Let the width of the river be D.

$Vnet=Vb+Vr\\Vnet=V+4V\\Vnet=5V.$

The width of the river will be,

$Speed=\frac{Distance}{Time} \\Distance=SpeedXTime\\D=VnetX1\\D=5VX1\\D=5V.$

Second case: When only the speed of the river is operational and the speed of the motorboat got reduced.

Let us assume that the speed of the motorboat got reduced by a factor n.

Speed of the motorboat=$\frac{4V}{n}$.

$Vnet=Vb+Vr\\Vnet=V+\frac{4V}{n} \\Vnet=\frac{nV+4V}{n} .$

The width of the river will be,

$Speed=\frac{Distance}{Time} \\Distance=SpeedXTime\\D=VnetX4\\D=(\frac{nV+4V}{n} )X4.$

D will still be the same so, the value of n can be found by,

$5V=(\frac{nV+4V}{n})X4\\5V=\frac{4nV+16V}{n} \\5nV=4nV+16V\\5nV-4nV=16V\\nV=16V\\n=16.$

Hence, the motorboat's speed got reduced by a factor 16.

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