Answer:
a) The speed of the father is 3.38 m/s
b) The speed of the son is 8.28 m/s
Explanation:
Hi there!
The equation of kinetic energy is as follows:
KE = 1/2 · m · v²
Where:
KE = kinetic energy
m = mass
v = speed
The kinetic energy of the father will be:
KEf = 1/2 · mf · vf²
Where the letter "f" indicates "father"
The kinetic energy of the son will be:
KEs = 1/2 · ms · vs²
KEs = 1/2 · (1/3 · mf) · vs²
Where the letter "s" indicates "son".
We know that the kinetic energy of the father is half the kinetic energy of the son. Then:
KEf = 1/2 KEs
1/2 · mf · vf² = 1/2 · (1/2 · 1/3 · mf · vs²)
Let´s solve this equation for vf:
1/2 · mf · vf² = 1/12 · mf · vs²
Multiply by 2 and divide by mf both sides of the equation:
vf² = 1/6 · vs²
6 · vf² = vs²
We also know that when the father speeds up by 1.4 m/s the kinetic energy of the son and father is equal. Then:
KEf = 1/2 · mf · (vf + 1.4 m/s)²
KEs = 1/2 · 1/3 · mf · vs²
KEf = KEs
1/2 · mf · (vf + 1.4 m/s)² = 1/2 · 1/3 · mf · vs²
divide both sides of the equation by mf and 1/2
(vf + 1.4 m/s)² = 1/3 · vs²
(vf + 1.4 m/s) · (vf + 1.4 m/s) = 1/3 · vs²
vf² +2.8 vf + 1.96 = 1/3 · vs²
Replace vs² by 6 vf²
vf² +2.8 vf + 1.96 = 1/3 · 6 vf²
vf² +2.8 vf + 1.96 = 2 vf²
subtract 2 vf to both sides of the equation:
-2 vf² + vf² +2.8 vf + 1.96 = 0
-1 vf² + 2.8 vf + 1.96 = 0
Let´s solve the quadratic equation using the quadratic formula:
a = -1
b = 2.8
c = 1.96
vf = [-b ± √(b² - 4ac)]/2a
vf = 3.38 m/s
(The other solution of the quadratic equation is negative and therefore discarded).
The speed of the son will be:
6 · vf² = vs²
6 · (3.38 m/s)² = vs²
vs = 8.28 m/s
Then, the velocity of the son is 8.28 m/s and the velocity of the father is 3.38 m/s
