A golfer on a level fairway hits a ball at an angle of 38° to the horizontal that travels 85 yd before striking the ground. He then hits another ball from the same spot with the same speed, but at a different angle. This ball also travels 85 yd. At what angle was the second ball hit? (Neglect air resistance.)

To find the angle at which the second ball was hit, we can use the concept of projectile motion.

In projectile motion, the horizontal and vertical components of the motion are independent of each other. The horizontal component represents the distance covered, while the vertical component determines the height or elevation.

Let's break down the problem and find the initial velocity of the first ball.

Given:
Angle of projection (θ1) = 38°
Distance traveled (d) = 85 yd

We are assuming that the initial velocity (v1) remains constant for both shots.

The horizontal component of the initial velocity is given by v1 * cos(θ1), and the vertical component is v1 * sin(θ1).

Using trigonometric identities:
Horizontal component = v1 * cos(θ1) = d
Vertical component = v1 * sin(θ1)

Since we know the distance traveled, we can solve for the initial velocity:

v1 * cos(θ1) = d
v1 = d / cos(θ1)

Now, for the second shot, we need to find the angle θ2 at which the ball was hit. We know the initial velocity is the same (v1), and the distance traveled is also the same (d).

Similar to the first shot, we can find the horizontal and vertical components:

Horizontal component = v1 * cos(θ2) = d
Vertical component = v1 * sin(θ2)

Dividing these two equations, we can eliminate v1:

(v1 * sin(θ2)) / (v1 * cos(θ2)) = sin(θ2) / cos(θ2) = tan(θ2)

Therefore, from the trigonometric identity:
tan(θ2) = sin(θ2) / cos(θ2)

Since we know that the two distances and initial velocities are equal, we can substitute them in the equation:

tan(θ2) = sin(θ1) / cos(θ1)

Finally, to find θ2, we can take the arctan of both sides:

θ2 = arctan(sin(θ1) / cos(θ1))

Plugging in the given values:
θ2 = arctan(sin(38°) / cos(38°))

Using a calculator, we can find the value of θ2 to be approximately 52.7°.

Therefore, the second ball was hit at an angle of approximately 52.7°.