Power — Question 6

Question 6

Drag makes steady-speed power grow faster than force

A vehicle has mechanical drive power 16 kW. Its only resistance on a level road is $F_d=cv^2$ opposite motion, where $c=2$ N s$^2$/m$^2$. Assume drive power can remain constant at the speeds considered.

1. Find the steady speed.

2. Find mechanical power required to maintain 30 m/s.

3. At 10 m/s, is kinetic energy increasing or decreasing, and at what rate?

Question 6 - Solution

Drag makes steady-speed power grow faster than force

(a) Balance powers at steady speed.

The governing relation is $$\boxed{\frac{dK}{dt}=P_{\mathrm{drive}}-F_dv=P_{\mathrm{drive}}-cv^3}$$

$$P_d=F_dv=cv^3.$$ Thus $$16000=2v^3,$$ giving $$v=20\ \mathrm{m/s}.$$

(b) Apply the cubic dependence.

$$P(30)=2(30)^3=54000\ \mathrm W,$$ or 54 kW. Increasing speed by a factor 1.5 increases required power by $$1.5^3=3.375.$$

(c) Subtract resistance power from drive power.

At 10 m/s, resistance removes $$2(10)^3=2000\ \mathrm W.$$ Net power is $$16000-2000=14000\ \mathrm W,$$ so kinetic energy increases at 14 kJ/s. Mass would be needed to determine acceleration.