Answer:
a) , b)
, c)
,
.
Step-by-step explanation:
a) The function in terms of time and the inital angle measured from the horizontal is:
The particular expression for the cannonball is:
b) The components of the position of the cannonball before hitting the ground is:
After a quick substitution and some algebraic and trigonometric handling, the following expression is found:
The angle for a maximum horizontal distance is determined by deriving the function, equalizing the resulting formula to zero and finding the angle:
Now, it is required to demonstrate that critical point leads to a maximum. The second derivative is:
Which demonstrates the existence of the maximum associated with the critical point found before.
c) The equation for the vertical component of position is:
The maximum height can be found by deriving the previous expression, which is equalized to zero and critical values are found afterwards:
Now, the second derivative is used to check if such solution leads to a maximum:
Which demonstrates the assumption.
The maximum height reached by the cannonball is: