Let's solve this problem step-by-step by determining the kinetic energy of the moving body first and then using that energy to find the height for the potential energy.
Step 1: Calculate the Kinetic Energy (KE) of the Moving Body
The formula for kinetic energy is given by:
[tex]\[ \text{KE} = \frac{1}{2} m v^2 \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the body (in kg)
- [tex]\( v \)[/tex] is the velocity of the body (in m/s)
Given:
- Mass ([tex]\( m \)[/tex]) = 10 kg
- Velocity ([tex]\( v \)[/tex]) = 10 m/s
Substituting these values into the formula:
[tex]\[ \text{KE} = \frac{1}{2} \times 10 \times (10)^2 \][/tex]
[tex]\[ \text{KE} = \frac{1}{2} \times 10 \times 100 \][/tex]
[tex]\[ \text{KE} = 5 \times 100 \][/tex]
[tex]\[ \text{KE} = 500 \, \text{Joules} \][/tex]
So, the kinetic energy of the moving body is 500 Joules.
Step 2: Determine the Height for Equivalent Potential Energy (PE)
The formula for potential energy is given by:
[tex]\[ \text{PE} = mgh \][/tex]
where:
- [tex]\( m \)[/tex] is the mass of the body (in kg)
- [tex]\( g \)[/tex] is the acceleration due to gravity (approximately [tex]\( 9.8 \, \text{m/s}^2 \)[/tex])
- [tex]\( h \)[/tex] is the height above the ground (in meters)
We need to find the height [tex]\( h \)[/tex] such that the potential energy [tex]\( \text{PE} \)[/tex] is equal to the kinetic energy [tex]\( \text{KE} \)[/tex].
Given:
- Mass ([tex]\( m \)[/tex]) = 10 kg
- Gravitational acceleration ([tex]\( g \)[/tex]) = 9.8 m/s[tex]\(^2\)[/tex]
- Potential energy ([tex]\( \text{PE} \)[/tex]) = 500 Joules
Substituting these values into the potential energy formula and solving for height [tex]\( h \)[/tex]:
[tex]\[ 500 = 10 \times 9.8 \times h \][/tex]
[tex]\[ 500 = 98h \][/tex]
[tex]\[ h = \frac{500}{98} \][/tex]
[tex]\[ h \approx 5.102 \, \text{m} \][/tex]
So, the height at which a body of mass 10 kg must be situated to have potential energy equal to the kinetic energy of another body of the same mass moving with a velocity of 10 m/s is approximately 5.102 m.
Therefore, the correct answer is approximately 5 m, which corresponds to option (a).
