A father racing his son has one-third the kinetic energy of the son, who has two-fifths the mass of the father. The father speeds up by 2.5 m/s and then has the same kinetic energy as the son.
Sam,
By reposting 4 times (plus the original) in a matter of minutes, you are not helping the system.
Either someone has to spend the time deleting your duplicate posts instead of making a response (which is the present case), or all physics tutors have to spend time reading the same post over 5 times, thus reducing the time spent on responding.
To solve this problem, let's break it down step by step:
Step 1: Define the variables:
Let F be the mass of the father.
Let S be the mass of the son.
Let Kf be the initial kinetic energy of the father.
Let Ks be the initial kinetic energy of the son.
Let Kff be the final kinetic energy of the father (after he speeds up).
Let Kfs be the final kinetic energy of the son.
Step 2: Write the given information as equations:
According to the problem, the following information is given:
1. Kf = 1/3 * Ks (The initial kinetic energy of the father is one-third the kinetic energy of the son.)
2. S = 2/5 * F (The mass of the son is two-fifths the mass of the father.)
Step 3: Write the equations for kinetic energy:
The kinetic energy (K) of an object can be calculated using the equation:
K = (1/2) * m * v^2
where K is the kinetic energy, m is the mass of the object, and v is the velocity of the object.
Using this equation, we can write expressions for the kinetic energy of the father (Kf) and the son (Ks) in terms of their respective masses and velocities.
Kf = (1/2) * F * v_f^2
Ks = (1/2) * S * v_s^2
Step 4: Substitute the given information into the equations:
Substituting the given information from Step 1 and Step 2 into the equations in Step 3:
Kf = (1/2) * F * v_f^2
Ks = (1/2) * S * v_s^2
Substituting Kf = 1/3 * Ks:
(1/3 * Ks) = (1/2) * F * v_f^2
Substituting S = 2/5 * s = (1/2) * (2/5 * F) * v_s^2
Step 5: Simplify the equations:
Now, let's simplify the equations obtained in Step 4.
Equation 1: (1/3 * Ks) = (1/2) * F * v_f^2
Multiplying both sides by 3 to eliminate the fraction:
Ks = (3/2) * F * v_f^2
Equation 2: Ks = (1/2) * (2/5 * F) * v_s^2
Simplifying the expression by multiplying the fractions:
Ks = (1/5) * F * v_s^2
Step 6: Set the final kinetic energy equal for both father and son:
According to the problem, after the father speeds up, his kinetic energy becomes equal to the son's kinetic energy. Let's denote the final kinetic energy of the father as Kff.
Kff = (1/5) * F * (v_f + 2.5)^2
Step 7: Set up the final equation:
We can now set up an equation by equating the final kinetic energy of the father (Kff) and the son (Kfs):
Kff = Kfs
Substituting the expressions obtained in Step 5 and the equation obtained in Step 6:
(1/5) * F * (v_f + 2.5)^2 = (3/2) * F * v_f^2
Step 8: Solve the equation:
Now, solve the equation from Step 7 to find the value of v_f:
(1/5) * (v_f + 2.5)^2 = (3/2) * v_f^2
Simplifying the equation and moving terms to one side:
(1/10) * (v_f^2 + 5v_f + 6.25) = (3/2) * v_f^2
Multiplying both sides by 10 to eliminate fractions:
v_f^2 + 5v_f + 6.25 = 15v_f^2
Bringing like terms on one side:
15v_f^2 - v_f^2 - 5v_f - 6.25 = 0
Combining like terms:
14v_f^2 - 5v_f - 6.25 = 0
Step 9: Solve the quadratic equation:
Now, solve the quadratic equation obtained in Step 8 to find the value(s) of v_f.
Use the quadratic formula:
v_f = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values a = 14, b = -5, and c = -6.25 into the quadratic formula:
v_f = (-(-5) ± √((-5)^2 - 4 * 14 * -6.25)) / (2 * 14)
Simplifying the equation:
v_f = (5 ± √(25 + 350)) / 28
v_f = (5 ± √375) / 28
Now, calculate the two possible values of v_f by evaluating both the positive and negative square root:
v_f ≈ 2.33 m/s or v_f ≈ -0.27 m/s
Since velocity cannot be negative in this context, the father's final velocity (v_f) after speeding up is approximately 2.33 m/s.