How Did the Father Match His Son's Kinetic Energy After Speeding Up?

[netrunnr]

A father racing his son has half the kinetic energy of the son who has half the mass of the father. The father speeds up by 1.0m/s and then has the same kinetic energy as the son. what are the original speeds of the father and the son?
using k=1/2mv^2 and solving for v I did this:

M = mass K = kinetic energy V=velocity
d = (dad) father s = son i = initial f=final

Initial father
Md
Kdi
Vd

Initial son
Ms = 1/2 Md
Ks = 2Kdi
Vs

Kd = 1/2 Md (Vd )^2
Ks = 1/2 1/2 Md (Vd) ^2
2(1/2 Md (Vd)^2 = 1/2 1/2 Md Vs^2
2(2Vd^2 = 1/2 Vs^2
4Vd^2 = 1Vs^2

Final father
Md
Kdf
Vd+1.0ms
Final Son
Ms = 1/2 Md
Ks = Kdf
Vs

Kd = 1/2 Md (Vd +1)^2
Ks = 1/2 1/2 Md (V2)^2
1/2Md ( Vd +1)^2 = 1/2 1/2 Md Vs^2
now taking from initial the 4Vd^2 = 1Vs^2
(Vd +1)^2 = 1/2Vs^2
(Vd +1) ^2 = 1/2 4Vd^2
Vd^2 +2Vd + 1 = 2Vd^2
Vd^2 + 2Vd + 1 -2Vd^2 = 0
-Vd^2 +2Vd + 1 =0

I was trying to solve it at this point like a quadratic equation but I am lost because it seems unsolvable. I know I made a mistake and can't see where...

 

[alphysicist]

Hi netrunnr,

A father racing his son has half the kinetic energy of the son who has half the mass of the father. The father speeds up by 1.0m/s and then has the same kinetic energy as the son. what are the original speeds of the father and the son?
using k=1/2mv^2 and solving for v I did this:

M = mass K = kinetic energy V=velocity
d = (dad) father s = son i = initial f=final

Initial father
Md
Kdi
Vd

Initial son
Ms = 1/2 Md
Ks = 2Kdi
Vs

Kd = 1/2 Md (Vd )^2
Ks = 1/2 1/2 Md (Vd) ^2

This is a typo; it should be Vs. (But it looks like you correct it in the next line.)

2(1/2 Md (Vd)^2 = 1/2 1/2 Md Vs^2
2(2Vd^2 = 1/2 Vs^2

I don't believe this is correct, but it is probably just a typo because the next line is correct.

4Vd^2 = 1Vs^2

This line is correct, and so once you find the initial speed of the dad Vd, you can find the speed of the son.

Final father
Md
Kdf
Vd+1.0ms
Final Son
Ms = 1/2 Md
Ks = Kdf
Vs

Kd = 1/2 Md (Vd +1)^2
Ks = 1/2 1/2 Md (V2)^2
1/2Md ( Vd +1)^2 = 1/2 1/2 Md Vs^2
now taking from initial the 4Vd^2 = 1Vs^2
(Vd +1)^2 = 1/2Vs^2
(Vd +1) ^2 = 1/2 4Vd^2
Vd^2 +2Vd + 1 = 2Vd^2
Vd^2 + 2Vd + 1 -2Vd^2 = 0
-Vd^2 +2Vd + 1 =0

I was trying to solve it at this point like a quadratic equation but I am lost because it seems unsolvable. I know I made a mistake and can't see where...

It looks right to me. Just put it in the quadratic equation and solve. What do you get?