Answer:
The change in power is 4400 W.
Explanation:
Given that,
Power = 10 kW
Speed = 10 m/s
Increases speed = 12 m/s
Given equation is,
[tex]F=kv[/tex]
We know that,
The power is,
[tex]P=Fv[/tex]
Put the value of F into the formula
[tex]P=(kv)v[/tex]
[tex]P=kv^2[/tex]
[tex]P\propto v^2[/tex]
We need to calculate the new power
Using formula for power
[tex]\dfrac{P}{P'}=\dfrac{v^2}{v'^2}[/tex]
Put the value into the formula
[tex]\dfrac{10}{P'}=(\dfrac{10}{12})^2[/tex]
[tex]P'=(\dfrac{12}{10})^2\times10[/tex]
[tex]P'=14.4\ kW[/tex]
We need to calculate the change in power
Using formula of change in power
[tex]\Delta P=P'-P[/tex]
Put the value into the formula
[tex]\Delta P=14.4-10[/tex]
[tex]\Delta P=4.4\ kW[/tex]
[tex]\Delta P=4.4\times1000[/tex]
[tex]\Delta P=4400\ W[/tex]
Hence, The change in power is 4400 W.
This question involves the concepts of Power, velocity, and force.
The increase in power needed for the boat to move through the lake at a constant speed of 12 m/s is "c. 4400 W".
The power in terms of velocity and force is given by the following formula:
[tex]P = Fv[/tex]
where,
P = Power
F = Force
v = velocity
But, the force can be given by the following formula, as stated in the question:
[tex]F=kv\\\\Therefore,\\\\P=(kv)v\\P=kv^2\\\\k=\frac{P}{v^2}\\\\Hence,\\\\\frac{P_1}{v_1^2}=\frac{P_2}{v_2^2}\\\\[/tex]
where,
Pâ = Initial Power = 10 KW = 10000 W
Pâ = Final Power = ?
vâ = Initial Velocity = 10 m/s
vâ = Final Velocity = 12 m/s
Therefore,
[tex]\frac{10000\ W}{(10\ m/s)^2}=\frac{P_2}{(12\ m/s)^2}[/tex]
Pâ = 14400 W
Now, the increase in power will be:
ÎP = Pâ - Pâ = 14400 W - 10000 W
ÎP = 4400 W
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