Kinetic energy of runner race question

[OVB]

Say a father who has a mass that is two times that of his son is racing against him, and his kinetic energy is half of his son. When the father increases his speed by one m/s, the kinetic energies are equal.

I do this:
M = mass of father
V = velocity of father
0.5MV^2 = 0.5(0.5mv^2)

2MV^2 = mv^2
(4m)V^2 = mv^2

4V^2 = v^2

2V = v

0.5M(V+1)^2 = 0.5m(2V)^2
(V+1)^2 = 4V^2
V^2 + 2V + 1 = 4V^2

-3V^2 +2V + 1 = 0

(3V +1) (-V + 1)

V = 1, -1/3

so V = 1 m/s

However, my book says the speeds are 2.4 m/s and 4/8 m/s for father and son, respectively. What am I doing wrong?

 

[semc]

M & V =mass and velocity of father right?
m & v =mass and velocity of son?
since the father initial KE is half his son, why did u multiply .5 on the KE of the son instead of the father?

btw,my ans for velocity of son is 3.414m/s

 

[OVB]

No, that is how it should be. KE of F = 0.5(KE of son)

Does anyone know why the answers are 2.4 and 4.8?

 

[OlderDan]

Say a father who has a mass that is two times that of his son is racing against him, and his kinetic energy is half of his son. When the father increases his speed by one m/s, the kinetic energies are equal.

I do this:
M = mass of father
V = velocity of father
0.5MV^2 = 0.5(0.5mv^2)

2MV^2 = mv^2
(4m)V^2 = mv^2

4V^2 = v^2

2V = v

0.5M(V+1)^2 = 0.5m(2V)^2
2m(V+1)^2 = m(2V)^2 <== added
(V+1)^2 = 2V^2
V^2 + 2V + 1 = 2V^2

-1V^2 +2V + 1 = 0

etc.

(3V +1) (-V + 1)

V = 1, -1/3

so V = 1 m/s
However, my book says the speeds are 2.4 m/s and 4/8 m/s for father and son, respectively. What am I doing wrong?

See the colors

 

[petergel]

The first thing we do is relate the father's kinetic energy to the son's according to the question. I will keep the father on the LHS and son on the RHS. I will used lowercase v for the father's velocity and uppercase V for the son's velocity.

1) (0.5)(m)(v*v) = (0.5)(0.5)(0.5m)(V*V)
// Now multiply by 8 to remove fraction...
4m(v*v) = m(V*V)
// Now divide by m to simplify...
4(v*v) = (V*V)
// Now take square root of both sides.
2v = V
// This gives us the son's velovity V in terms of the
// father's velocity v. ie: V = 2v.

Now in order to have the father's K equal the son's K we do two things.
- Add 1 to the father's velocity on the LHS.
- Multiply the RHS by 2 since we are not relating the father's K to
half the son's K anymore. ie: Instead of K = 0.5K we now have
K = K since that's what happens when we add 1 to the father's
velocity.

2) (0.5)(m)(v+1)(v+1) = (0.5)(0.5m)(2v)(2v)
// Remember V = 2v
// Multiply by 2 and divide by m to simplify...
(v+1)(v+1) = (2)(v*v)
// Take the square root of both sides...
v+1 = sqrt(2)*v
v = 1 / (sqrt(2) - 1)
v = 2.41 m/s.

Now all we have to do is substitute into V = 2v to get the son's original
velocity...
V = 2 * 2.41 = 4.82 m/s.

 

[Rocket_guy]

you went wrong on this part:

0.5M(V+1)^2 = 0.5m(2V)^2
(V+1)^2 = 4V^2
V^2 + 2V + 1 = 4V^2

-3V^2 +2V + 1 = 0

(3V +1) (-V + 1)

V = 1, -1/3

so V = 1 m/s

0.5M(V+1)^2 = 0.5m(2V)^2 ... subtitute M=2m here. this will give you ..
2(V+1)^2 = 4V^2
V^2 + 2V + 1 = 2V^2

-V^2 +2V + 1 = 0
0r V^2-2V-1=0
solving I get V= 2.4142 m/sec.

use this to calculate v=2V=2*2.4142=4.8284 meters per second.