A stone is thrown horizontally with speed 30m/s from a tower of 50 m ,height of stone from ground at which it make 45' with horizontally is

To determine the height at which the stone makes a 45-degree angle with the horizontal, we need to analyze the stone's motion both vertically and horizontally. When the stone is thrown horizontally from a height, its vertical motion is influenced by gravity while it maintains a constant horizontal velocity.

Analyzing the Motion

First, let's break down the stone's horizontal and vertical movements:

Horizontal Motion

The stone has an initial horizontal velocity of 30 m/s. Since there is no horizontal acceleration (ignoring air resistance), the horizontal distance covered by the stone can be described by the formula:

• Horizontal Distance (x) = Velocity (v) × Time (t)

Vertical Motion

The stone is dropped from a height of 50 meters. The vertical motion is influenced by gravity, which accelerates the stone downwards at approximately 9.81 m/s². The vertical distance fallen can be described by the equation:

• Vertical Distance (y) = Initial Height (h) - (1/2)gt²

Here, g is the acceleration due to gravity (9.81 m/s²), and t is the time in seconds.

Finding the Height at 45 Degrees

At a 45-degree angle, the vertical and horizontal distances traveled by the stone are equal. Thus, when the stone makes a 45-degree angle with the horizontal, we can say:

• x = y

Setting Up the Equations

From the horizontal motion, we can express time t in terms of horizontal distance:

• x = 30 m/s × t
• Therefore, t = x / 30

Substituting t into the vertical motion equation gives:

• y = h - (1/2)g(t²)
• y = 50 - (1/2)(9.81)(x/30)²

Equating Horizontal and Vertical Distances

Since at a 45-degree angle, x = y, we can set up the equation:

• x = 50 - (1/2)(9.81)(x/30)²

This simplifies to a quadratic equation in terms of x. Solving this equation will give us the horizontal distance x at which the stone makes a 45-degree angle.

Calculating the Values

To simplify calculations, let's denote the term involving gravity:

• g/2 = 4.905 m/s²

The equation then becomes:

• x = 50 - 4.905(x/30)²

To find the roots of this equation, let's rearrange it:

• 4.905(x/30)² + x - 50 = 0

Now, substituting values and solving the quadratic equation would yield the value of x. Once we obtain x, we can substitute back to find y, which will give us the height above the ground when the stone makes a 45-degree angle with the horizontal.

Final Thoughts

After performing these calculations, you will find the specific height at which the stone will achieve the desired angle. Keep in mind that understanding the interplay between horizontal and vertical motions is key in projectile motion problems.