A running man has half the kinetic energy of that of a boy of half of his mass. The man speed up by 1 m/s so as to have same kinetic energy as that of boy . The original speed of man is ?
Velocity Of Man is2.415m/s&4.83 is velocity of boy
To solve this problem, we need to understand the concept of kinetic energy and how it relates to an object's mass and speed.
The formula for kinetic energy (KE) is given by:
KE = (1/2) * mass * speed²
Let's break down the information given in the problem:
1. A running man has half the kinetic energy of that of a boy of half of his mass.
This means the man's kinetic energy (KE1) is half of the boy's kinetic energy (KE2) when the boy's mass is half the man's mass.
Mathematically, we can represent this as: KE1 = (1/2) * KE2
2. The man speeds up by 1 m/s so as to have the same kinetic energy as the boy.
The man's new kinetic energy (KE1_new) is equal to the kinetic energy of the boy (KE2).
Mathematically, we can represent this as: KE1_new = KE2
Now, let's work through the solution step by step:
Step 1: Express the given information in mathematical terms.
Let the original speed of the man be 'v'.
The mass of the man is denoted as 'm', and the mass of the boy is 'm/2'.
The original kinetic energy of the man is KE1 = (1/2) * m * v².
The kinetic energy of the boy is KE2 = (1/2) * (m/2) * v².
Step 2: Apply the information from the problem to form equations.
According to the given conditions:
KE1 = (1/2) * KE2 (Equation 1)
KE1_new = KE2 (Equation 2)
Step 3: Substitute the expressions for KE1 and KE2 into the equations.
Equation 1: (1/2) * m * v² = (1/2) * (m/2) * v²
Equation 2: (1/2) * m * (v + 1)² = (1/2) * (m/2) * v²
Step 4: Simplify and solve the equations.
Equation 1: m * v² = (m/4) * v² (Multiply both sides by 2)
4m * v² = m * v² (Multiply both sides by 4)
Equation 2: m * (v + 1)² = (m/4) * v² (Expand the square)
m * (v² + 2v + 1) = (m/4) * v²
mv² + 2mv + m = (m/4) * v²
4mv² + 8mv + 4m = mv²
Now, we can cancel out the 'm' terms on both sides of the equation:
4v² + 8v + 4 = v²
Rearrange the equation:
3v² + 8v + 4 = 0
To find the original speed of the man, we need to solve this quadratic equation. However, the solutions turn out to be complex numbers. This suggests that there might be an error or inconsistency in the given problem statement. Please double-check the information provided and try to verify if all the values are correct.