A stone is thrown upwards from the roof with a velocity 15ms^-1 at an angle 30degree to the horizontal. The height of the building is 40m.calculate the magnitude of the velocity of the stones before it strikes the ground.
Yo = 15*sin30 = 7.5 m/s. = Ver. component of initial velocity.
Y^2 = Yo^2 + 2g*h = 0.
h = -Yo^2/2g = -(7.5)^2/-19.6 = 2.87 m.
above the roof.
V^2 = Vo^2 + 2g*(h+40) = 0 +19.6*42.87 = 840.25
V = 29 m/s.
To calculate the magnitude of the velocity of the stone before it strikes the ground, we'll need to break down the initial velocity into its horizontal and vertical components.
Given:
Initial velocity (u) = 15 m/s
Angle of projection (θ) = 30 degrees
Height of the building (h) = 40 m
Acceleration due to gravity (g) = 9.8 m/s² (assuming no air resistance)
First, let's find the vertical component of the initial velocity (u_y). This can be calculated using the following equation:
u_y = u * sin(θ)
Substituting the given values:
u_y = 15 m/s * sin(30°)
Next, we'll calculate the time taken for the stone to reach the ground. Since the stone is being thrown upwards, it will take the same time to reach its peak height as it does to come back down. We can use the following equation to find the time of flight (T):
T = 2 * u_y / g
Substituting the values:
T = 2 * (15 m/s * sin(30°)) / 9.8 m/s²
Now, we can use the time of flight (T) to calculate the vertical distance traveled (s) using the following formula:
s = u_y * T - (1/2) * g * T²
Substituting the values:
s = (15 m/s * sin(30°)) * [(2 * (15 m/s * sin(30°)) / 9.8 m/s²)] - (1/2) * 9.8 m/s² * [(2 * (15 m/s * sin(30°)) / 9.8 m/s²)]²
Simplifying the equation, we find:
s = 40 m
The stone will reach a maximum height of 40 meters before coming back down to the ground.
Now, let's calculate the total time of flight (T_total). We know that the total distance traveled vertically (s_total) is equal to the height of the building (h) plus the distance traveled to reach the peak height (h_peak). Thus:
s_total = h + h_peak
h_peak = s_total - h
h_peak = 40 m - 40 m = 0 m
The stone reaches its peak height of 0 meters, and we can conclude that it does not go higher than the starting point on the roof of the building. Therefore, the total time of flight (T_total) is equal to the time to reach the peak height.
Now, to find the magnitude of the velocity of the stone before it strikes the ground (v_f), we'll use the equation:
v_f = u_y + g * T_total
Substituting the values:
v_f = (15 m/s * sin(30°)) + 9.8 m/s² * [(2 * (15 m/s * sin(30°)) / 9.8 m/s²)]
Simplifying the equation, we find:
v_f = 15 m/s
Therefore, the magnitude of the velocity of the stone before it strikes the ground is 15 m/s.