To determine how fast a boy of 40 kg mass should run so that his kinetic energy becomes 800 Joules, let's break down the solution step by step.
### Step 1: Understanding Kinetic Energy
The kinetic energy ([tex]\(KE\)[/tex]) of a moving object is given by the formula:
[tex]\[ KE = \frac{1}{2} \times \text{mass} \times \text{velocity}^2 \][/tex]
Where:
- [tex]\( KE \)[/tex] is the kinetic energy in Joules (J)
- [tex]\( \text{mass} \)[/tex] is the mass of the object in kilograms (kg)
- [tex]\( \text{velocity} \)[/tex] is the velocity of the object in meters per second (m/s)
### Step 2: Rearrange the Formula to Solve for Velocity
To find the velocity ([tex]\(\text{velocity}\)[/tex]), we need to rearrange the kinetic energy formula. Start by isolating [tex]\(\text{velocity}\)[/tex]:
[tex]\[ KE = \frac{1}{2} \times \text{mass} \times \text{velocity}^2 \][/tex]
Multiply both sides by 2 to eliminate the fraction:
[tex]\[ 2 \times KE = \text{mass} \times \text{velocity}^2 \][/tex]
Now, divide both sides by the mass:
[tex]\[ \text{velocity}^2 = \frac{2 \times KE}{\text{mass}} \][/tex]
Finally, take the square root of both sides to solve for [tex]\(\text{velocity}\)[/tex]:
[tex]\[ \text{velocity} = \sqrt{\frac{2 \times KE}{\text{mass}}} \][/tex]
### Step 3: Substitute the Given Values
Now, we substitute the given values into the equation:
- Mass ([tex]\( \text{mass} \)[/tex]) = 40 kg
- Kinetic Energy ([tex]\( KE \)[/tex]) = 800 J
[tex]\[ \text{velocity} = \sqrt{\frac{2 \times 800}{40}} \][/tex]
### Step 4: Calculate the Expression Inside the Square Root
First, calculate the numerator:
[tex]\[ 2 \times 800 = 1600 \][/tex]
Next, divide by the mass:
[tex]\[ \frac{1600}{40} = 40 \][/tex]
### Step 5: Take the Square Root
Now, take the square root of 40:
[tex]\[ \text{velocity} = \sqrt{40} \][/tex]
The computed value of the square root of 40 is:
[tex]\[ \text{velocity} \approx 6.324 \, \text{m/s} \][/tex]
### Conclusion
The boy should run at a speed of approximately 6.324 meters per second (m/s) to have a kinetic energy of 800 Joules given his mass of 40 kilograms.
