a paricle projected upwards from the top of tower with an initial velocity 40 m/s and angle of projection 30° with the horizontal falls at a distance of 200√3 from foot of tower. The height of tower is (g= 10m/s²

To determine the height of the tower from which the particle is projected, we can break down the problem into two main components: the horizontal motion and the vertical motion of the particle. By analyzing these components separately, we can find the height of the tower.

Understanding the Motion Components

The initial velocity of the particle is given as 40 m/s at an angle of 30° with the horizontal. We can resolve this initial velocity into its horizontal and vertical components using trigonometric functions:

• Horizontal Component (Vx): Vx = V * cos(θ) = 40 * cos(30°) = 40 * (√3/2) = 20√3 m/s
• Vertical Component (Vy): Vy = V * sin(θ) = 40 * sin(30°) = 40 * (1/2) = 20 m/s

Calculating Time of Flight

The horizontal distance traveled by the particle is given as 200√3 meters. We can use the horizontal motion equation to find the time of flight (t):

Distance = Horizontal Velocity × Time

200√3 = 20√3 × t

Solving for t:

t = 200√3 / 20√3 = 10 seconds

Vertical Motion Analysis

Next, we need to analyze the vertical motion to find the height of the tower. The vertical displacement (h) can be calculated using the following kinematic equation:

h = Vy * t - (1/2) * g * t²

Substituting the known values:

• Vy = 20 m/s
• g = 10 m/s²
• t = 10 s

Now, plugging in these values:

h = 20 * 10 - (1/2) * 10 * (10)²

h = 200 - (1/2) * 10 * 100

h = 200 - 500

h = -300 m

Interpreting the Result

The negative height indicates that the particle falls below the level of the tower after reaching its peak. This means the particle was projected from a height of 300 meters above the ground. Therefore, the height of the tower is:

Height of the tower = 300 meters

In summary, by breaking down the motion into horizontal and vertical components and applying the kinematic equations, we were able to determine that the height of the tower is 300 meters. This approach illustrates the importance of analyzing projectile motion in a systematic way, allowing us to solve for unknowns effectively.