Power Functions
ExplanationThe power rule helps find the derivative for the power terms inside of power functions. A power term consists of a coefficient, a variable, and an exponent. The process for using the power function is fairly simple: multiply the the coefficient by the exponent and then subtract one from the exponent. Some things that one should be aware of the derivative of a cubic function is a squared function, the derivative of a squared function is linear, and the derivative of a linear equation is constant.
|
Power Rule |
Simple Problems |
Word Problems
Since we know our rate of acceleration constant, we know that our velocity can be modled by the equation V = K * t2, where K is a constant and t is time.
If the train reaches its maximum speed of 320km/h in 15 minutes, then working backwards, we can write (320) = K * (15m)2, which simplifies to (320) = K * 225. And then 320/225= K
1.422222222=K. Our formula for velocity is V = 1.422222222km/h * t2 on the interval of 0 ≤ t ≤ 15
The formula for Force is F=M*A where M=mass in kg and A=acceleration in m/s.
To find the rate of acceleration, you take the derivative of Velocity
Derivative
V = 1.422222222km/h * t2
(1.42222222*2)*t2-1
2.84444444km/ht=A
Since we need A in m/s...
2.844444444km/h= 2844.44444m/60s = 47.4074074m/s*t=A
3 Minutes
F=M*A
F=(8*70625)*(47.4074074*t)
F=565000*47.4074074*(3)
F=80355555.543 newtons.
F=80.355555543 mega-newtons.
13 Minutes
F=M*A
F=(8*70625)*(47.4074074*t)
F=565000*47.4074074*(13)
F=348207407.35 newtons.
F=348.20740735 mega-newtons
20 Minutes
F=M*A
F=(8*70625)*(47.4074074*t)
F=565000*47.4074074*(20)
F=535703703.62 newtons.
F=535.70370362 mega-newtons
The force for 3, and 13 minutes are realistic. The Force for 20 minutes however is not. Because the train stops accelerating once it reaches 320 km/h, which happens in 15 minutes, the acceleration at 20 minutes is actually 0. At that point, the train is moving because of newtons law "an object in motion remains in motion", but is not actually exuding any Force to move at all.