A golfer hits a ball off of the ground with an angle of elevation of 10 degrees and an initial velocity of 240 feet per second. There is an 80 foot tree 160 yards in front of her. Will she hit the ball over the tree?

hf=hi-1/2 g t^2

so what is t?
t=horizontal distance/horizonal velocity

solve for hf.

To determine whether the golfer will hit the ball over the tree, we need to calculate the maximum height reached by the ball and compare it to the height of the tree.

First, let's convert the initial velocity from feet per second to yards per second because the distance to the tree is given in yards. There are 3 feet in a yard, so 240 feet per second is equal to (240/3) yards per second, which is 80 yards per second.

To find the maximum height reached by the ball, we can use the kinematic equation for vertical displacement:

y = yo + vyo * t - (1/2) * g * t^2

where:
y = vertical displacement (maximum height)
yo = initial vertical position (0 since the ball is on the ground)
vyo = initial vertical velocity (the vertical component of the initial velocity)
g = acceleration due to gravity (approximately 32.2 feet per second^2)

Since the angle of elevation is 10 degrees, the vertical component of the initial velocity can be found using the equation:

vyo = v * sin(theta)

where:
v = magnitude of the initial velocity
theta = angle of elevation

In this case, v = 80 yards per second and theta = 10 degrees.

So, vyo = 80 * sin(10)

Now we can substitute the values into the first equation to find the maximum height:

y = 0 + (80 * sin(10)) * t - (1/2) * 32.2 * t^2

To find the time it takes for the ball to reach its maximum height, we set the vertical velocity equal to zero:

vy = vyo - g * t = 0

Solving for t:

t = vyo / g

Now we can substitute this value of t back into the equation for y to get the maximum height:

y = 0 + (80 * sin(10)) * (vyo / g) - (1/2) * 32.2 * (vyo / g)^2

Finally, we compare the maximum height to the height of the tree, which is given as 80 feet or (80/3) yards. If the maximum height is greater than the height of the tree, then the golfer will hit the ball over the tree.

So, we need to calculate the maximum height and compare it to (80/3) yards.