A father racing his son has 1/3 the kinetic energy of the son, who has 1/2 the mass of the father. The father speeds up by 1.4 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

Answer: The initial velocity of the father is 1.91m/s

and the one of the son is 4.68 m/s

Explanation:

First, the kinetic energy is written as:

K = (m/2) * v^2

where m is mass and v is velocity.

The information that we have is that:

If we use tabs (M, K, V) as the data for the father and (m, k, v) as the data for the son:

K = k/3

m = M/2

now, let's proced with these two equations:

the first one says:

M*V^2 = (m*v^2) / 3

(where i cancelled the 1/2 in both sides)

now, we can replace the second equation into this and get:

M*V^2 = ((M/2) * v^2) / 3 = (M*v^2) / 6

Here we can divide by M in both sides, and we get:

V^2 = (v^2) / 6

We apply the square root to both sides, and we get:

V = v / (√6)

Afther that, the father increases his velcity by 1.4m/s and then the kinetic energys are equal, so now we have:

M * (V + 1.4) ^2 = m*v^2

again, using that m = M/2

M * (V + 1.4m/s) ^2 = (M/2) * v^2

We divide by M in both sides:

(V + 1.4m/s) ^2 = (v^2) / 2

Now we apply the square root to both sides:

V + 1.4m/s = v/√2

Now we have two equations and two variables:

V + 1.4m/s = v/√2

V = v / (√6)

We can replace the second equation in the first one and obtain the velocity of the son:

v / (√6) + 1.4m/s = v/√2

v (1/√6 - 1/√2) = - 1.4m/s

v = 1.4m/s / (-1/√6 + 1/√2) = 1.4m/s*0.299 = 4.68m/s

Now, knowing the velocity of the son, we can find the velocity of the father using:

V = v / (√6) = (4.68m/s) / √6 = 1.91m/s

Vf = 1.91m/s. Vs = 4.67m/s

Explanation:

Given that that Ms = 1/2 * Mf and that

(K. E) f = 1/3 * (K. E) s

Then

1/2 * Mf * Vf² = 1/3 * Ms*Vs²

1/2 * Mf*Vf² = 1/3 * 1/2 * Mf*Vs²

Dividing out common coefficients

Vf² = Vs²/6

Vs² = 6Vf²

The father increases his speed by 1.4m/s

Thereby having the same kinetic energy as the son. That is

1/2*Mf * (Vf + 1.4) ² = 1/2*Ms*Vs²

Dividing through by 1/2 and sybstituting for Ms and Vs

Mf * (Vf² + 2.8Vf + 1.96) = 1/2*Mf*6Vf²

Dividing through by Mf

Vf² + 2.8Vf + 1.96 = 3Vf²

Rearranging,

Vf² - 2.8Vf - 1.96 = 0

Solving the equation above by a calculator

Vf = 1.91m/s.

Substituting this value into the the equation for Vs,

Vs² = 6Vf² = 6 * 1.91² = 21.89

Taking the square root

Vs = 4.67m/s