A stone is projected up with a vertical velocity u, reaches upto a maximum height h. When it is at a height of 3h/4 from the ground, the ratio of KE and PE at that point is: (consider PE=0 at the point of projection)
a]1:1 b]1:2 c]1:3 d]3:1
To find the ratio of kinetic energy (KE) and potential energy (PE) at a given point, we first need to understand the concept of energy conservation.
At any point during the stone's motion, the total mechanical energy (TME) remains constant, which is the sum of the kinetic energy and potential energy. Mathematically, TME = KE + PE.
Initially, when the stone is projected upwards, it has only kinetic energy and no potential energy. Therefore, at the point of projection, the initial kinetic energy (KEi) is equal to the total mechanical energy (TMEi) while the potential energy (PEi) is zero. So, we have KEi = TMEi and PEi = 0.
When the stone reaches its maximum height h, it momentarily comes to a stop. At this point, it has zero kinetic energy (KEf) and the entire mechanical energy (TMEf) is in the form of potential energy (PEf). Thus, KEf = 0 and PEf = TMEf.
Now, let's consider the point when the stone is at a height of 3h/4 from the ground. At this point, we can calculate the ratio of KE and PE.
Since mechanical energy is conserved, we have TMEi = TMEf.
TMEi = KEi + PEi = KEf + PEf
TMEi = KEf + PEf
Since KEi = TMEi and KEf = 0, we can rewrite the equation as:
KEi = KEf + PEf
Substituting the values, we have:
TMEi = 0 + PEf
TMEi = PEf
So, the total mechanical energy at the point when the stone is at a height of 3h/4 from the ground is equal to the potential energy.
Since the potential energy is equal to the total mechanical energy, the ratio of KE:PE at that point is 0:1 or simply 0:1.
Therefore, the correct answer is d) 3:1.
To find the ratio of kinetic energy (KE) to potential energy (PE) when the stone is at a height of 3h/4 from the ground, we need to understand the concept of energy conservation and the formulas for kinetic and potential energy.
First, let's analyze the situation. The stone is projected upward with a vertical velocity u. It reaches a maximum height h and then starts coming down. We are interested in the ratio of KE to PE when the stone is at a height of 3h/4 from the ground.
At the point of projection, the potential energy is taken to be zero. As the stone reaches the maximum height h, all the initial kinetic energy gets converted into potential energy. At this point, the stone momentarily comes to rest before falling back down. Therefore, at the maximum height, the potential energy is maximum, and the kinetic energy is zero.
Now, let's consider the stone when it is at a height of 3h/4 from the ground. We can use the principle of energy conservation to determine the ratio of KE to PE at this point. Energy conservation states that the total mechanical energy (sum of kinetic and potential energy) of an object remains constant in the absence of external forces.
Since there are no external forces acting on the stone except gravity, we can say that the total mechanical energy at any point is equal to the total mechanical energy at the point of projection.
Now let's calculate the ratio of KE to PE at a height of 3h/4 from the ground.
Since the stone is at a height of 3h/4 from the ground, the potential energy at this point is 3/4 of the maximum potential energy, which is equal to 3/4 of PE at the maximum height.
The ratio of KE to PE can be calculated as follows:
KE/PE = (Total mechanical energy at the point of projection) / (Potential energy at 3h/4)
Since the total mechanical energy at the point of projection is equal to PE + KE = PE + 0 (as there is no kinetic energy at the point of projection), we can substitute PE for the total mechanical energy at the point of projection in the ratio equation:
KE/PE = PE / (Potential energy at 3h/4)
KE/PE = PE / (3/4 * PE)
Simplifying the ratio equation:
KE/PE = 4/3
Therefore, the ratio of KE to PE at a height of 3h/4 from the ground is 4:3.
So, the correct option is c) 1:3.
Let u = Vo = 10 m/s.
g = 10 m/s^2.
hmax=(V^2-Vo^2)/2g = (0-100)/-20 = 5 m.
(3/4)hmax = (3/4)*5 = 3.75 m.
V^2 = Vo^2 + 2g*h
V^2 = 100 - 20*3.75 = 25
V = 5 m/s. @ (3/4)hmax.
When h = 0,
KE + PE = 0.5M*V^2 + 0 = 0.5M*V^2 =
0.5M*10^2 = 50M. = KE.
M = Mass.
When h = (3/4)hmax,
KE + PE = 50M
0.5M*5^2 + PE = 50M
PE == 50M - 12.5M = 37.5M
KE/PE = 12.5M/37.5M = 1/3 or 1:3.