A body is projected vertically upward and the height hmeters reached after a time t seconds is given by h=196t-4.9t^2.find the time taken to reach the greatest height and the greatest height reached
To find the time taken to reach the greatest height, we need to find the vertex of the equation h = 196t - 4.9t^2.
The vertex of a parabola in the form y = ax^2 + bx + c is given by the x-coordinate: x = -b/2a.
In this case, the equation is h = -4.9t^2 + 196t.
Comparing this to the general form y = ax^2 + bx + c, we have:
a = -4.9
b = 196
Using the formula for the x-coordinate of the vertex, we calculate:
t = -196 / (2*(-4.9))
t = -196 / -9.8
t = 20
So, the time taken to reach the greatest height is 20 seconds.
To find the greatest height reached, we substitute this time back into the equation:
h = 196(20) - 4.9(20)^2
h = 3920 - 4.9(400)
h = 3920 - 1960
h = 1960
Therefore, the greatest height reached is 1960 meters.
To find the time taken to reach the greatest height, we need to determine when the velocity of the body becomes zero. The maximum height is reached at this point, where the velocity changes from positive to negative.
Let's differentiate the height equation with respect to time to calculate the velocity function:
v = dh/dt = d/dt (196t - 4.9t^2)
= 196 - 9.8t
To find the time taken to reach the greatest height, we set the velocity equal to zero and solve for t:
0 = 196 - 9.8t
Rearranging the equation, we get:
9.8t = 196
t = 196 / 9.8
t ≈ 20 seconds
Therefore, it takes approximately 20 seconds for the body to reach the greatest height.
To find the greatest height reached, substitute the value of t = 20 back into the original height equation:
h = 196t - 4.9t^2
h = 196(20) - 4.9(20)^2
h = 3920 - 4.9(400)
h = 3920 - 1960
h = 1960 meters
Hence, the greatest height reached by the body is 1960 meters.