A father racing his son has 1/3 the kinetic energy of the son, who has 1/4 the mass of the father. The father speeds up by 1.8 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?
Son : m, v
Father : M, u
M=4m
mv²/2=3Mu²/2 =>
mv²/2=3•4mu²/2 =>
v²=12 u²…..(1)
mv²/2= M(u+1.8)²/2
mv²/2= 4m(u+1.8)²/2
v²= 4(u+1.8)² ….(2)
Substitute v² from (1) in (2)
12 u²=4(u+1.8)²
3u²=u²+3.6u+3.24
u²-1.8u-1.62 = 0
u=2.46 m/s
v=sqrt(12 u²) = 3.46u = 8.52 m/s
To solve this problem, we'll use the equations for kinetic energy and the relationship between mass and kinetic energy. As given, the father initially has 1/3 the kinetic energy of the son.
Let's denote:
- Mass of the father as M.
- Mass of the son as m.
- Initial velocity of the father as v_f.
- Initial velocity of the son as v_s.
We are given the following information:
1. The father has 1/3 the kinetic energy of the son: 1/3 * (1/2) * M * v_f^2 = (1/2) * m * v_s^2
2. The son has 1/4 the mass of the father: m = (1/4) * M
To solve for the original velocities, we'll follow these steps:
Step 1: Simplify the equations and rewrite them in terms of one variable.
Using the given information (equation 2), substitute m in equation 1:
1/3 * (1/2) * M * v_f^2 = (1/2) * [(1/4) * M] * v_s^2
Simplifying further:
1/6 * M * v_f^2 = 1/8 * M * v_s^2
Step 2: Cancel out common factors.
Divide both sides by (1/6 * M):
v_f^2 = (1/8) * v_s^2
Step 3: Solve for v_s in terms of v_f.
Rearrange the equation:
v_s^2 = 8 * v_f^2
Taking the square root of both sides:
v_s = 2 * sqrt(2) * v_f
Step 4: Substitute the new expression for v_s into equation 2 to find the relationship between the velocities.
m = (1/4) * M
(1/4) * M = (1/4) * M * 4 * (2 * sqrt(2) * v_f)^2
Simplifying:
1 = 8 * v_f^2
Step 5: Solve for v_f.
Divide both sides by 8:
v_f^2 = 1/8
Taking the square root of both sides:
v_f = 1/2
(a) The original speed of the father (v_f) is 1/2 m/s.
Step 6: Substitute the value of v_f into the expression for v_s:
v_s = 2 * sqrt(2) * v_f
v_s = 2 * sqrt(2) * (1/2)
Simplifying:
v_s = sqrt(2)
(b) The original speed of the son (v_s) is sqrt(2) m/s.
So, the original speeds were:
(a) The father's speed was 1/2 m/s.
(b) The son's speed was sqrt(2) m/s.