运动包含二部分:
1.速度从0,以a1做匀加速运动,达到速度V,用时间为t1,路程s1
s1=1/2a1t1^2 , v=a1t1
2.速度从V,以a2的加速度减速运动到停车,所用时间t2,路程s2
s2=vt2-1/2 a2t2^2 ,v=a2t2
最短时间:T=t1+t2 S=s1+s2
v=a1t1=a2t2=a2(T-t1)==>t1=a2T/(a1+a2), t2=a1t1/a2
s2=vt2-1/2 a2t2^2
=(a1t1)(a1t1/a2)-1/2 a2(a1t1/a2)^2
=(a1t1)^2/a2-1/2(a1t1)^2/a2
=1/2(a1t1)^2/a2
S=s1+s2=1/2a1t1^2+1/2(a1t1)^2/a2
=1/2a1(a1+a2)t1^2/a2
t1=a2T/(a1+a2),
可以求出 T=√2S(a1+a2)/a1a2
V=a1t1=a1a2T/(a1+a2)=√2Sa1a2/(a1+a2)