a stone is thrown vertically upwards from the ground to reach a maximum height of 90m .how long will it take to reach a height of 75m
well, it gets to 75 twice,once on the way up and once on the way down.
first find initial speed up, Vi
(1/2) m Vi*2= m g (90)
Vi = sqrt (2*9.81*90) = 42.0 m/s
then
v = 42 - .81 t
h = 0 + 42 t - 4.9 t^2 = 75
4.9 t^2 - 42 t + 75 = 0
https://www.mathsisfun.com/quadratic-equation-solver.html
V^2 = Vo^2 + 2g*h = 0.
Vo^2-19.6*90 = 0
Vo = 42 m/s.
V^2 = Vo^2+2g*h = 42^2+(-19.6)75 = 294
V = 17.1 m/s. at 75 m.
V = Vo+g*T = 17.1
42-9.8T = 17.1
T = 2.5 s. to reach 75 m.
To determine how long it will take for the stone to reach a height of 75m, we need to calculate the time it takes for the stone to reach its maximum height and then come back down to a height of 75m.
Let's start by finding the time it takes for the stone to reach its maximum height of 90m. We can use the equation:
h = ut + (1/2)gt^2
Where:
h = maximum height (90m)
u = initial velocity (when thrown vertically upwards, the initial velocity is 0 m/s)
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time
Plugging in the known values:
90 = 0t + (1/2)(9.8)(t^2)
Simplifying the equation:
90 = 4.9t^2
Dividing both sides by 4.9:
18.3673 = t^2
Taking the square root of both sides:
t ≈ 4.28 seconds
So, it takes approximately 4.28 seconds for the stone to reach its maximum height.
Now, let's calculate the time it takes for the stone to come back down from the maximum height to a height of 75m. Since the stone is thrown vertically upwards, the time it takes to come back down will be the same as the time it took to reach the maximum height.
Therefore, it will take approximately 4.28 seconds for the stone to reach a height of 75m.
To find the time it takes for the stone to reach a height of 75m, we can use the principles of projectile motion.
In projectile motion, the key parameter when object moves vertically upward is the initial velocity. The initial velocity when the stone is thrown upwards is the same as when it returns back to the ground, but with opposite sign.
First, let's determine the initial velocity.
We know that when the stone reaches the maximum height of 90m, its velocity will be zero. At the maximum height, the stone will momentarily come to rest before falling back down due to gravity. Therefore, we can set up an equation using the kinematic equation for vertical motion:
vf = vi + gt,
where
vf = final velocity (which is 0 m/s at the maximum height)
vi = initial velocity
g = acceleration due to gravity (-9.8 m/s^2, since the stone is moving upwards)
Plugging in the values, we get:
0 = vi - 9.8t,
where t is the time it takes for the stone to reach the maximum height.
Simplifying the equation, we have:
vi = 9.8t.
Next, we can use this initial velocity to determine the time it takes for the stone to reach a height of 75m.
Using the same kinematic equation for vertical motion:
h = vi*t + 0.5*g*t^2,
where
h = height
vi = initial velocity
g = acceleration due to gravity
t = time
Plugging in the values, we get:
75 = (9.8t)*t + 0.5*(-9.8)t^2.
Simplifying the equation, we have:
75 = 9.8t^2 - 4.9t^2.
Combining like terms, we have:
4.9t^2 = 75.
To solve for t, divide both sides of the equation by 4.9:
t^2 = 75/4.9.
Taking the square root of both sides, we have:
t = √(75/4.9).
Using a calculator to evaluate the square root, we find:
t ≈ 3.08 seconds.
Therefore, it will take approximately 3.08 seconds for the stone to reach a height of 75m.