Power — Question 5

Question 5

A force cap and a power cap control different acceleration stages

A 1000 kg vehicle starts from rest on a level frictionless track. Its drive can exert at most 2000 N and deliver at most 20 kW. It always uses the largest force allowed by both limits.

1. Find the speed at which the governing limit changes.

2. Find the time to reach that speed.

3. Find the additional time to reach 20 m/s and the total time.

Question 5 - Solution

A force cap and a power cap control different acceleration stages

(a) Set the two force limits equal.

The governing relation is $$\boxed{F(v)=\min\!\left(F_{\max},\frac{P_{\max}}v\right)\quad(v>0)}$$

$$F_{\max}=P_{\max}/v_c,$$ giving $$v_c=20000/2000=10\ \mathrm{m/s}.$$

(b) Use constant acceleration below the transition.

$$a=2000/1000=2\ \mathrm{m/s^2}.$$ The first stage takes $$10/2=5\ \mathrm s.$$ Its power rises from zero to 20 kW.

(c) Use constant energy rate above the transition.

$$\Delta K=\tfrac12(1000)(20^2-10^2)=150000\ \mathrm J.$$ The second stage takes $$150000/20000=7.5\ \mathrm s,$$ so total time is 12.5 s. Applying constant power from rest would violate the force cap.