If an object is fired upward from the top of a tower at a velocity of 80 feet per second, the tower is 200 ft high, the formula is
h(t)=-16t^2+80t+200, t is time h is height, how long after it is fired does the object reach the max height? It is like a quadratic equation
If an object is fired upward from the top of a tower at a velocity of 80 feet per second, the tower is 200 ft high, the formula is
h(t)=-16t^2+80t+200, t is time h is height, how long after it is fired does the object reach the max height? It is like a quadratic equation
The classic equation for height reached by a body projected upward is h = Vot - gt^2/2 or Vot - 16t^2 where Vo = the initial upward velocity, t = the time of flight and g = the acceleration due to gravity.
Your equation of height reached is given by h = 200 + 80t - 16t^2.
The equation of motion from which you can obtain the time of flight is Vf = Vo - gt where Vf = the final velocity. Since the final final velocity is zero, we have 0 = 80 - 32t making t = 2.5 sec.
After 2.5 sec., it will have reached its maximum height of 200 + 80(2.5) - 16(2.5)^2 = 200 + 200 - 100 = 300 ft.
Well, well, well, looks like we've got ourselves a height-seeking missile here! To find out when this little projectile reaches its maximum height, we need to find the peak of the quadratic equation h(t) = -16t^2 + 80t + 200.
Now, let me tell you a little secret: the maximum height occurs when the object is at its peak, which means its velocity is zero. So, we gotta find out when this thing goes from going up to coming back down.
To do that, we take the derivative of h(t) with respect to t and set it equal to zero. Now hold onto your hat, because things are about to get mathematical!
h'(t) = -32t + 80
Setting h'(t) equal to zero and solving for t, we get:
-32t + 80 = 0
-32t = -80
t = 2.5 seconds
So, after a lengthy calculation session, we find out that this object reaches its maximum height 2.5 seconds after it is fired. Hope you enjoyed the ride, it was a real high-flying adventure!
To find the time at which the object reaches its maximum height, we need to determine the vertex of the quadratic equation h(t) = -16t^2 + 80t + 200. The x-coordinate of the vertex corresponds to the time at which the object reaches the maximum height.
The formula for finding the x-coordinate of the vertex of a quadratic equation in the form ax^2 + bx + c = 0 is given by x = -b / (2a).
In this equation, a = -16 and b = 80. Plugging these values into the formula, we get:
x = -80 / (2 * (-16))
x = -80 / (-32)
x = 2.5
Therefore, the object reaches its maximum height 2.5 seconds after it is fired.
To determine when the object reaches its maximum height, we need to find the time at which the height function, h(t), is maximized.
Since the height function is a quadratic equation, we can find the maximum point by using the formula for the x-coordinate of the vertex, given by:
t = -b / (2a)
In the equation h(t) = -16t^2 + 80t + 200, the coefficient of t^2 is -16, and the coefficient of t is 80. Plugging these values into the formula, we get:
t = -80 / (2 * (-16))
t = -80 / (-32)
t = 2.5
Therefore, the object reaches its maximum height 2.5 seconds after it is fired upward from the top of the tower.