An object is fired vertically upward with an initial speed of 68.6 meters/second. After t seconds, the height is shown by
y = -4.9 t^2 + 68.6 t meters. Given that this object has an initial speed of 68.6 m/s, what is the maximum height it will reach? How long did it take the rocket to reach the maximum height?
Using calculus, it would be finding dy/dt and equating to zero to solve for t.
Since it is algebra, you could complete squares:
y=-4.9(t-7)²+49*4.9
=-4.9(t-7)²+240.1
which means that the maximum height of 240.1 m at t=7.
To find the maximum height the object will reach, we need to determine the vertex of the parabolic equation representing the height.
The equation given for the height is:
y = -4.9 t^2 + 68.6 t
This equation represents a downward-opening parabola since the coefficient of the t^2 term is negative. The vertex of a parabolic equation represents the maximum point of the curve.
The vertex of a parabolic equation in the form y = ax^2 + bx + c is given by:
x = -b / (2a)
In our case, a = -4.9 and b = 68.6. Substituting these values into the formula, we find:
t = -68.6 / (2*(-4.9))
t = -68.6 / (-9.8)
t ≈ 7
So, it takes approximately 7 seconds for the object to reach its maximum height.
To find the maximum height, substitute the time of the maximum point (t = 7) into the equation for the height:
y = -4.9(7^2) + 68.6(7)
y ≈ -4.9(49) + 480.2
y ≈ -240.1 + 480.2
y ≈ 240.1
Therefore, the maximum height the object will reach is approximately 240.1 meters.