A fully charged battery in a solar-powered car of mass m delivers a constant
thrust of F Newtons. The rolling resistance of the car is σv where v is its
velocity in m/s, and σ is a known coefficient of friction. The air resistance is
ρv², where ρ is a drag coefficient. These resistance forces are in the opposite
direction to the thrust as shown in the diagram.

(a) Use Force = mdv/dt to write down a differential equation satisfied by the
velocity v.

(b) Suppose F = 100 N, m = 70 kg, σ = 3 kg/s, and ρ = 1/10 kg/m.
i. Find the general solution of the differential equation satisfied by
v. You will need to use partial fractions.

ii. Find the solution for the velocity of the car if it drives at full
thrust starting from rest.

iii. Plot the solution as a function of time.

iv. Show that the top speed cannot exceed 72 km/h.

打不了希腊字母，用p代替rho, 用q代替sigma.

(1) F-pv^2-qv=mdv/dt

(2) 如果F=100 (N), m=70(kg),q=3 (kg/s),p=1/10(kg/m)
那么方程化为
100-1/10v^2-3v=70dv/dt
700dv/(v^2+30v-1000)=-dt
700dv/[(v-20)(v+50)]=-dt
700*(1/70)*(1/(v-20)-1/(v+50))dv=-dt两边积分
10*(ln(v-20)-ln(v+50))=-t+C1
ln[(v-20)/(v+50)]=-t/10+C2
(v-20)/(v+50)=C*e^(-t/10)
v=(50C*e^(-t/10)+20)/(1-C*e^(-t/10))
这就是微分方程的general solution。

t=0, v=0, -->(50C+20)/(1-C)=0, C=-20/50
所以这个时候速度为
v=1000*(1-e^(-t/10))/(50+20e^(-t/10))

自己画图。

v=1000*(1-e^(-t/10))/(50+20e^(-t/10))

这是个关于t的单调增加函数，所以当t-->infinity， v-->1000/50=20.速度不能超过20m/s=1200m/min=72000m/h=72km/h。