In: Physics
A cannon will launch balls at different angles but with the same inital velocity magnitude. If the cannon sits on the edge of a cliff that has a slope of 60 degrees, what is the distance along the slope of the ledge that the ball travels? If the inital velocity was 30 m/s and at which angle would the maximum distance be achieved?
The situation can be visualized by
The initial vertical velocity is
The acceleration in the vertical direction is
The distance traveled in the vertical direction is
The equation of motion in the vertical direction is
The horizontal velocity throughout the motion is (there is no acceleration in the horizontal direction)
The equation of motion in the horizontal direction is
Putting this value in the first equation
Now from the diagram
Putting these values
Now, let us put the values of the constants in our problem
Now initially
So, the length along the slope is given by
So, the answer to the first part is 636.26 m.
Now, to maximize the value of l, we need to take the first derivative of the function and put it equal to zero.
The solution to this equation is
So, the maximum range along the slope is at 15o launch angle.