A running man has half the kinetic energy of a boy half his mass. The man speeds up by 1.30 m/s and then has the same kinetic energy as the boy. What is the speed (in meters/second) of the boy?
man velocity: x
boy velocity: b
1/2 (2m)x^2 = 1/2 * 1/2 my^2
4x^2 = y^2
1/2 (2m)(x+1.3)^2 = 1/2 my^2
2(x+1.3)^2 = 4x^2
x^2 + 2.6x + 1.69 = 2x^2
x^2 - 2.6x - 1.69 = 0
x = 3.138
check:
1/2 (2m)*3.138^2 = 1/2 * 1/2 m*6.276^2
9.847 = 1/2 * 19.694 = 9.847
1/2 (2m)*4.438^2 = 1/2 m*6.276^2
19.694 = 19.694
To solve this problem, we can use the equation for kinetic energy:
Kinetic Energy (KE) = 1/2 * mass * velocity^2
Let's define the variables:
- Let M be the mass of the running man.
- Let V1 be the initial velocity of the running man.
- Let V2 be the final velocity of the running man (V2 = V1 + 1.30 m/s).
- Let m be the mass of the boy.
- Let Vb be the velocity of the boy.
The given information states that the running man has half the kinetic energy of the boy when their masses are halved. Mathematically, this can be represented as:
(1/2) * M * V1^2 = (1/2) * (m/2) * Vb^2
Simplifying this equation, we get:
M * V1^2 = (1/4) * m * Vb^2
Next, we are told that when the running man increases his velocity by 1.30 m/s, he will have the same kinetic energy as the boy. Mathematically:
M * V2^2 = m * Vb^2
Substituting V2 = V1 + 1.30 into the equation, we get:
M * (V1 + 1.30)^2 = m * Vb^2
We now have two equations:
1) M * V1^2 = (1/4) * m * Vb^2
2) M * (V1 + 1.30)^2 = m * Vb^2
To find the velocity of the boy (Vb), we need to solve this system of equations. Let's solve them simultaneously:
First, divide equation 2 by equation 1:
(V1 + 1.30)^2 / V1^2 = 4
Simplifying this equation, you will get:
(V1 + 1.30)^2 = 4 * V1^2
Expand the binomial and rearrange:
V1^2 + 2.60 * V1 + 1.69 = 4 * V1^2
Combining like terms:
3 * V1^2 - 2.60 * V1 - 1.69 = 0
Now, we have a quadratic equation. We can solve this equation using the quadratic formula:
V1 = (-b ± √(b^2 - 4ac)) / 2a
Substituting the values into the quadratic formula, we get:
V1 = (-(-2.60) ± √((-2.60)^2 - 4*3*(-1.69))) / (2*3)
Simplifying this equation will give us two possible values for V1, which we will call V1_1 and V1_2.
Using each value of V1 in equation 1, we can find the corresponding value of Vb.
Finally, the speed of the boy (Vb) will be the positive solution since it represents the physical velocity.