An object is fired vertically upward with an initial speed of 68.6 meters/second.
After t seconds, the height is shown by y= -4.9t^2+68.6t 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?
For the first part, use conservation of energy, and let Vo be the initial veloicty
(Vo^2)/2 = gH
Solve for maximum height H
H = 240 m
Time to reach maximum height
= H/V(average) = 240/34.3 = 7.43 s
To find the maximum height, we need to determine the vertex of the quadratic equation y = -4.9t^2 + 68.6t. The vertex of a quadratic equation in the form y = ax^2 + bx + c is given by the equation t = -b / (2a).
In this case, the equation is y = -4.9t^2 + 68.6t, so a = -4.9 and b = 68.6.
Substituting the values into the equation for finding the vertex:
t = - (68.6) / (2 * -4.9)
t = -68.6 / -9.8
t ≈ 7
Therefore, it took approximately 7 seconds for the object to reach its maximum height.
To find the maximum height, substitute the value of t back into the equation for y:
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.
To find the maximum height the object will reach, we need to determine the vertex of the parabolic equation representing the height-time relationship.
The equation for the height as a function of time is given by y = -4.9t^2 + 68.6t.
We can rewrite this equation in the form y = a(t - h)^2 + k, where (h,k) represents the coordinates of the vertex. The coefficient "a" determines whether the parabola opens upward or downward. In this case, "a" is negative (-4.9), which means the parabola is concave downwards.
To find the time at which the rocket reaches the maximum height, we can use the formula t = -b/2a, where b represents the coefficient of the linear term.
In this case, the coefficient b is 68.6. Plugging the values into the formula, we have:
t = -68.6 / (2*(-4.9))
t = -68.6 / (-9.8)
t = 7
So, it takes the rocket 7 seconds to reach the maximum height.
Now, to find the maximum height, we substitute this time value back into the equation:
y = -4.9(7)^2 + 68.6(7)
y = -4.9(49) + 480.2
y = -240.1 + 480.2
y = 240.1
Thus, the maximum height the rocket will reach is 240.1 meters.