Afather racing his son has 1/4 the kinetic energy of the son, who has 1/3 the mass of the father. the father speeds up by 1.2 m/s and then has the same kinetic energy as the son. what are the original speeds of (a) the father and (b) the son?

Explanation:

KE_s: Kinetic Energy Son

KE_f: Kinetic Energy Father.

Relationship

KE_f: =  (1/4) KE_s

m_s: = (1/3) m_f

v_f: = velocity of father

v_s: = velocity of the son

Relationship

1/2 mf (v_f + 1.2)^2 = 1/2 m_s (v_s)^2      Multiply both sides by 2.

mf (v_f + 1.2)^2 = m_s * (v_s)^2               Substitute for the mass of the m_s

mf (v_f + 1.2)^2 = (m_f/3) * (v_s)^2         Divide both sides by father's mass

(v_f + 1.2)^2 = 1/3 * (v_s)^2                      multiply both sides by 3

3*(v_f + 1.2)^2 = (v_s) ^2                         Take the square root both sides

√3 * (v_f + 1.2) = v_s

Note

KE_f = (1/4) KE_s                                                  Multiply by 4

4* KE_f = KE_s                                                     Substitute (again)

4*(1/2) m_f (v_f + 1.2)^2 = 1/2* (1/3)m_f *v_s^2   Divide by m_f

2* (v_f + 1.2)^2 = 1/6 * (v_s)^2                              multiply by 6

12*(vf + 1.2)^2 = (v_s)^2                                        Take the square root

2*√(3* (v_f + 1.2)^2) = √(v_s^2)

2*√3 * (vf + 1.2) = v_s

Use the second relationship to substitute for v_s so you can solve for v_f

2*√3 * ( v_f + 1.2) = √3 * (v_f + 1.2)                     Divide by sqrt(3)

2(v_f + 1.2) = vf + 1.2

Edit

2vf + 2.4 = vf + 1.2

2vf - vf + 2.4 = 1.2

vf = 1.2 - 2.4

vf = - 1.2

This answer is not possible, but 2 of us are getting the same answer. The other person is someone whose math I would never question. She rarely makes an error. And I do mean rarely. Could you check to see that you have copied this correctly?

the total distance traveled by the car is

30km + 49km = 79 km .