A tower is 24 m high. A stone is projected upwards from the tower, at a velocitry of 24,5 m/s. Calculate how long it would take the stone to reach the ground at the roof of the tower.
Vf = Vo + g*Tr = 0.
24.5 + (-9.8)Tr = 0.
-9.8Tr = 24.5
Tr = 2.5 s. = Rise time.
Tf1 = Tr = 2.5 s = Time to fall back to roof.
h = Vo*t + 4.9*t^2 = 24 m.
24.5*t + 4.9t*2 = 24.
4.9t^2 + 24.5t -24 = 0.
Use Quad. Formula.
t = 0.839 s. = Time to fall from top of roof to gnd.
T = Tr + Tf1 + t = Tme to reach gnd.
To calculate how long it would take the stone to reach the ground at the roof of the tower, we need to consider two factors: the initial velocity of the stone and the height of the tower.
Given:
Initial velocity (u) = 24.5 m/s
Height of the tower (h) = 24 m
Using the equation of motion for free fall, specifically the equation for finding time (t), we can proceed step-by-step:
Step 1: Write down the equation of motion:
h = ut + (1/2)gt^2
Where:
h = height
u = initial velocity
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Step 2: Substitute the values into the equation:
24 = (24.5)t + (1/2)(9.8)t^2
Step 3: Rearrange the equation and simplify:
(1/2)(9.8)t^2 + (24.5)t - 24 = 0
Step 4: Solve the quadratic equation using the quadratic formula:
t = [-b ± √(b^2 - 4ac)] / (2a)
Where:
a = 0.5
b = 24.5
c = -24
Substituting the values:
t = [-24.5 ± √(24.5^2 - 4(0.5)(-24))] / (2(0.5))
Step 5: Calculate the time:
t = [-24.5 ± √(600.25 + 48)] / 1
t ≈ [-24.5 ± √648.25] / 1
Using a calculator, we find two possible values for t:
t ≈ 1.09 seconds (rounded to two decimal places)
or
t ≈ -24.59 seconds (rounded to two decimal places)
Since time cannot be negative in this context, we discard the negative value. Therefore, the stone would take approximately 1.09 seconds to reach the ground at the roof of the tower.
To find out the time it would take for the stone to reach the ground at the roof of the tower, we need to use the equations of motion. The key equation we can use here is:
h = ut + (1/2)at^2
Where:
h = height (in this case, 24 m)
u = initial velocity (24.5 m/s)
t = time taken
a = acceleration (in this case, acceleration due to gravity which is -9.8 m/s^2)
Since the stone is projected upwards, the acceleration due to gravity acts in the opposite direction, and therefore, we multiply it by -1 to make it negative.
Plugging in the values we know:
24 = (24.5)t + (1/2)(-9.8)t^2
Simplifying the equation:
0 = -4.9t^2 + 24.5t - 24
Now, we can solve this quadratic equation using either factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Applying the values from our quadratic equation:
t = [ -24.5 ± √((24.5)^2 - 4(-4.9)(-24)) ] / (2(-4.9))
Simplifying the equation further:
t = [ -24.5 ± √(600.25 - 470.4) ] / (-9.8)
t = [ -24.5 ± √(129.85) ] / (-9.8)
Calculating further:
t ≈ [ -24.5 ± 11.4 ] / (-9.8)
Now, we have two possible solutions for time:
1. When t = [-24.5 + 11.4] / (-9.8) ≈ 1.36 s
2. When t = [-24.5 - 11.4] / (-9.8) ≈ -3.05 s
Since time cannot be negative in this context, we discard the negative value.
Therefore, it would take approximately 1.36 seconds for the stone to reach the ground at the roof of the tower.