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Kinetic energy of runner race question

  1. Oct 22, 2006 #1


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    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?
  2. jcsd
  3. Oct 22, 2006 #2
    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:yuck:
    Last edited: Oct 22, 2006
  4. Oct 25, 2006 #3


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    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?
  5. Oct 26, 2006 #4


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    See the colors
  6. Jan 15, 2007 #5
    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

    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
    V = 2 * 2.41 = 4.82 m/s.
  7. Jan 15, 2007 #6
    you went wrong on this part:

    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.
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