A body projected vertically with a velocity “u” from the ground . It’s velocity at ¾ th of maximum height is

When a body is projected vertically upward with an initial velocity "u," it experiences a change in velocity due to the acceleration caused by gravity. To find the velocity of the body at three-quarters of its maximum height, we can break down the problem using the principles of kinematics and energy conservation.

Understanding the Motion

First, let's clarify a few key concepts. When an object is projected upwards, it will rise until it reaches its maximum height, where its velocity becomes zero before it starts to fall back down. The maximum height (H) can be calculated using the formula:

• H = (u²) / (2g)

Here, "g" represents the acceleration due to gravity, approximately 9.81 m/s². The maximum height is reached when the initial kinetic energy is completely converted into gravitational potential energy.

Calculating the Height at ¾ of Maximum Height

To find the velocity at three-quarters of the maximum height, we first determine what that height is:

• Height at ¾ of H = (3/4)H = (3/4) * (u² / (2g))

Using Energy Conservation

We can apply the principle of conservation of mechanical energy. At the initial point of projection, the total mechanical energy is purely kinetic:

• Initial Energy = (1/2)mu²

At three-quarters of the maximum height, the energy consists of both kinetic and potential energy:

• Potential Energy at ¾ H = mgh = mg * (3/4) * (u² / (2g)) = (3/8)mu²
• Kinetic Energy at ¾ H = (1/2)mv²

According to the conservation of energy:

• Initial Energy = Kinetic Energy + Potential Energy

Substituting the values we have:

• (1/2)mu² = (1/2)mv² + (3/8)mu²

Solving for Velocity

Now, we can simplify this equation. First, we can cancel "m" from all terms (assuming m is not zero):

• (1/2)u² = (1/2)v² + (3/8)u²

Next, we rearrange the equation to isolate v²:

• (1/2)v² = (1/2)u² - (3/8)u²

Finding a common denominator (which is 8) gives us:

• (1/2)v² = (4/8)u² - (3/8)u² = (1/8)u²

Multiplying both sides by 2 results in:

• v² = (1/4)u²

Taking the square root of both sides, we find:

• v = (1/2)u

Final Result

Thus, the velocity of the body at three-quarters of its maximum height is half of its initial velocity, or v = (1/2)u. This relationship highlights how the kinetic energy decreases as the body ascends, while potential energy increases until it reaches the peak of its trajectory.