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

The initial speed of the father is 1.9 m/s and the initial speed of the son is 4.65 m/s.

The given parameters;

The kinetic energy of the son is written as follows;

The kinetic energy of the father is written as follows;

When the fathers speed increases by 1.4 m/s;

The velocity of the son is calculated as follows;

Thus, the initial speed of the father is 1.9 m/s and the initial speed of the son is 4.65 m/s.

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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

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