Power — Question 4

Question 4

Constant power produces a square-root speed history

A 2 kg cart begins at speed 3 m/s and receives a constant 10 W of net mechanical power for 4 s on a horizontal track.

1. Find speed as a function of time and the final speed.

2. Find the driving force initially and finally, assuming it is parallel to velocity.

3. Explain why the same ideal constant-power force law cannot start smoothly from exact rest.

Question 4 - Solution

Constant power produces a square-root speed history

(a) Integrate the energy rate.

The governing relation is $$\boxed{\frac{dK}{dt}=P\Longrightarrow K(t)=K_0+Pt}$$

$$K(t)=K_0+Pt=9+10t\ \mathrm J.$$ Since $K=v^2$ here, $$v(t)=\sqrt{9+10t}\ \mathrm{m/s},\qquad v(4)=7\ \mathrm{m/s}.$$

(b) Divide power by instantaneous speed.

$F=P/v$. Initially $$F=10/3\ \mathrm N;$$ finally $$F=10/7\ \mathrm N.$$ Constant power does not require constant force or acceleration.

(c) Inspect the low-speed limit.

For fixed positive P, $F=P/v$ diverges as v approaches zero. A real drive must impose a finite force limit or ramp its power during startup; the given nonzero initial speed avoids the singularity.