One kind of slingshot consists of a pocket that holds a pebble and is whirled on a circle of radius r. The pebble is released from the circle at the angle θ so that it will hit the target. The angle in the drawing is 35.5°. The distance to the target from the center of the circle is d. (See the drawing below, which is not to scale.) The circular path is parallel to the ground, and the target lies in the plane of the circle. The distance d is nine times the radius r. Ignore the effect of gravity in pulling the stone downward after it is released and find the angle θ
To find the angle θ, we can start by considering the geometry of the situation.
Let's label the points on the diagram as follows:
- The center of the circle is labeled O.
- The point where the pebble is released is labeled A.
- The point where the pebble hits the target is labeled B.
Since the distance d is nine times the radius r, we have d = 9r.
Now, let's consider triangle OAB. This is a right triangle because the circular path is parallel to the ground. The angle θ is the angle between the hypotenuse OA and the horizontal axis.
We can use the tangent function to find θ. The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
In this case, the side opposite θ is the distance d, and the side adjacent to θ is the radius r. So we have:
tan(θ) = d / r
Substituting d = 9r, we get:
tan(θ) = 9r / r
tan(θ) = 9
To find θ, we can take the inverse tangent (or arctan) of both sides:
θ = arctan(9)
Using a calculator, we find that θ is approximately 83.6 degrees.