A 30-kg stone is dropped from a height of 100m. find its KE and PE when it is 50 m from the ground
MEf=MEi
KE=1/2mv^2
PE=mgh
mgh+1/2mv^2=mgh+1/2mv^2
Vi=0
So,
mgh=mgh+1/2mv^2
mgh=mgh+KE
(30*9.8*100)-(30*9.8*50)=KE
30(980-490)=KE
KE=30*490=14.7kj
PE@50m=KE@50m
To solve this problem, we can use the equations for gravitational potential energy (PE) and kinetic energy (KE). The formulas are as follows:
Potential Energy (PE) = mass (m) * gravitational acceleration (g) * height (h)
Kinetic Energy (KE) = 0.5 * mass (m) * velocity squared (v^2)
Given information:
Mass (m) = 30 kg
Initial height (h) = 100 m
Final height (h') = 50 m
To find the potential energy (PE) when the stone is 50 m from the ground:
PE = m * g * h'
1. Find the value of gravitational acceleration (g):
Gravitational acceleration (g) on Earth is approximately 9.8 m/s^2.
2. Calculate the potential energy (PE):
PE = m * g * h'
= 30 kg * 9.8 m/s^2 * 50 m
To find the kinetic energy (KE) when the stone is 50 m from the ground:
Kinetic energy (KE) at any height can be found using the conservation of mechanical energy principle. At any point during free fall, the total mechanical energy (PE + KE) remains constant.
3. Calculate the initial potential energy (PE) when the stone starts at a height of 100 m:
PE_initial = m * g * h (initial height)
= 30 kg * 9.8 m/s^2 * 100 m
4. Calculate the initial kinetic energy (KE):
Before the stone is dropped, it is not moving, so the initial kinetic energy is zero.
5. Calculate the sum of the initial potential and kinetic energies:
Total mechanical energy (E) at the top = PE_initial + KE_initial
6. Calculate the final kinetic energy (KE):
Using the conservation of mechanical energy, we can say that the total mechanical energy at the top is equal to the total mechanical energy when the stone is at a height of 50 m from the ground:
Total mechanical energy (E) at the top = KE_final + PE_final
Given that KE_initial = 0, we can rearrange the equation to solve for KE_final:
KE_final = Total mechanical energy (E) at the top - PE_final
7. Calculate the final potential energy (PE):
PE_final = m * g * h'
= 30 kg * 9.8 m/s^2 * 50 m
Now let's perform the calculations:
PE_final = 30 kg * 9.8 m/s^2 * 50 m
KE_final = Total mechanical energy (E) at the top - PE_final
Note: The Total mechanical energy (E) at the top is equal to the initial potential energy (PE_initial).
KE_final = PE_initial - PE_final
= (30 kg * 9.8 m/s^2 * 100 m) - (30 kg * 9.8 m/s^2 * 50 m)
To find the kinetic energy (KE) and potential energy (PE) of the stone when it is 50 meters from the ground, we need to understand the equations for calculating these energies.
Kinetic Energy (KE) formula:
KE = 1/2 * m * v^2
Potential Energy (PE) formula:
PE = m * g * h
Where:
m = mass of the object (30 kg in this case)
v = velocity of the object
g = acceleration due to gravity (approximately 9.8 m/s^2 on Earth)
h = height of the object above the ground
To calculate the KE and PE when the stone is 50 meters from the ground, we need to determine its velocity at that position. We can use the principle of conservation of energy.
Since the stone is dropped from a height of 100m, it will have only potential energy initially because it is not moving. Therefore, the initial KE is zero.
Initially, the stone's PE is given by:
PE_initial = m * g * h_initial
Using the values given:
m = 30 kg
g = 9.8 m/s^2
h_initial = 100 m
PE_initial = 30 kg * 9.8 m/s^2 * 100 m = 29400 J
At a height of 50 meters from the ground, half the initial height, the stone will have converted some of its potential energy into kinetic energy. At this point, the PE is:
PE_50m = m * g * h_50m
Using the same mass and acceleration due to gravity, we can find h_50m by subtracting the current height from the initial height:
h_50m = h_initial - h_50m = 100m - 50m = 50m
PE_50m = 30 kg * 9.8 m/s^2 * 50 m = 14700 J
To find the KE when the stone is 50 meters from the ground, we need to calculate the velocity at that position. We can use the equation for gravitational potential energy conversion to calculate the velocity:
PE_50m = KE + PE_50m
Solving for KE:
KE = PE_50m - PE_initial
Substituting the given values:
KE = 14700 J - 29400 J = -14700 J
Since the final result is negative, it means that the stone has lost energy and is not moving at this point. Therefore, its kinetic energy is zero when it is 50 meters from the ground.