A stone is projected upward at an angle of 60degree to the horizontal from the top of a tower of height 100meter and it hits the ground at a point. If the initial velocity of projection is 100meter. Calculate the maximum height of the stone above the ground. Calculate the time it takes to reach this height. Calculate the time of flight
<<If the initial velocity of projection is 100meter.>>
"100 m" does not have the proper dimension for a velocity. Do you mean 100 m/s? If so, the vertical component of Vo is
Vyo = Vo sin60 = 86.6 m/s
The stone will rise an additional distance H such that
gH = (Vyo)^2/2
H = 383 m.Max hweight above ground will be H + 100 = 483 m.
Time to reach max height = Vyo/g = 8.8 seconds.
For the total time of flight, add to that the time it takes to fall 483 m.
To calculate the maximum height of the stone above the ground, we can use the equations of projectile motion. Let's break down the given information:
- Initial velocity (u) = 100 m/s
- Launch angle (θ) = 60 degrees
- Height of the tower (h) = 100 meters
1. Maximum Height Calculation:
To calculate the maximum height (H) reached by the stone, we can use the equation:
H = (u^2 * sin^2(θ)) / (2 * g)
Where:
- u = initial velocity (100 m/s)
- θ = launch angle in radians (60 degrees = π/3 radians)
- g = acceleration due to gravity (approximately 9.8 m/s^2)
First, we need to convert the launch angle to radians:
θ_radians = θ * π / 180
θ_radians = (60 * π) / 180
θ_radians = π/3
Now we can calculate the maximum height (H):
H = (100^2 * sin^2(π/3)) / (2 * 9.8)
H = (100^2 * (sqrt(3)/2)^2) / (2 * 9.8)
H ≈ (100^2 * 3/4) / (2 * 9.8)
H ≈ (10000 * 3/4) / 19.6
H ≈ 75000 / 19.6
H ≈ 3816 meters
Therefore, the maximum height reached by the stone is approximately 3816 meters above the ground.
2. Time to Reach Maximum Height Calculation:
To calculate the time it takes to reach the maximum height, we can use the equation:
t = (u * sin(θ)) / g
Where:
- u = initial velocity (100 m/s)
- θ = launch angle in radians (π/3)
- g = acceleration due to gravity (approximately 9.8 m/s^2)
Now we can calculate the time (t):
t = (100 * sin(π/3)) / 9.8
t = (100 * (sqrt(3)/2)) / 9.8
t ≈ (100 * 0.866) / 9.8
t ≈ 86.6 / 9.8
t ≈ 8.84 seconds
Therefore, it takes approximately 8.84 seconds for the stone to reach the maximum height.
3. Time of Flight Calculation:
To calculate the total time of flight, we can use the equation:
T = 2 * t
Where:
- T = total time of flight
- t = time to reach maximum height (8.84 seconds)
Now we can calculate the total time of flight (T):
T = 2 * 8.84
T = 17.68 seconds
Therefore, the total time of flight is 17.68 seconds.