Question

A car is driven by a constant power p. If the car moves a distance x then the velocity v attained by the car is related to x as:
A) v*alpha*(x^(1/3))
B) v*alpha*root x
C) v*alpha*(x^(-1/2))
D) v*alpha*(x^(-1/3))

Answer 1

A car is driven by a constant power P. if the car moves a distance x then the velocity v attained by the car is related to x as :

(a) v ∝ $\sqrt[3]{x}$

(b) v ∝ $\sqrt{x}$

(c) v ∝ $\frac{1}{\sqrt{x}}$

(d) v ∝ $\frac{1}{\sqrt[3]{x}}$

answer : option (A) v ∝ $\sqrt[3]{x}$

here, car is drive by a constant power P.

i.e., P = constant (let K)

we know, the ratio of change of energy with respect to time is power.

⇒dE/dt = K

change in energy is known as work done.

i.e., E = W = F.x [ work done = force × displacement ]

⇒d(F.x)/dt = K

⇒F (dx/dt) = K

∵ F = ma = m(v dv/dx) , dx/dt = v

∴ m (v dv/dx) v = K

⇒m ∫v²dv = K∫dx

⇒mv³/3 = Kx

⇒v³ = 3Kx/m

⇒v = $\sqrt[3]{\frac{3K}{m}}\sqrt[3]{x}$

∴ v ∝ $\sqrt[3]{x}$

Therefore the correct option is (A).