Question

A ball thrown horizontally at vi = 30.0 m/s travels a horizontal distance of d = 55.0 m before hitting the ground. from what height h was the ball thrown?

Answer 1

To find the height from which the ball was thrown, we can use the equations of motion for projectile motion. The ball was thrown from a height of 44.1 meters.

Step-by-step explanation:

To find the height from which the ball was thrown, we can use the equations of motion for projectile motion. Since the ball is thrown horizontally, the initial vertical velocity is 0. We can use the equation h = (1/2) g t^2, where h is the height, g is the acceleration due to gravity, and t is the total time the ball is in the air.

Using the given values, we can substitute g = 9.8 m/s^2 and t = 3 s into the equation to solve for h: h = (1/2) * 9.8 m/s^2 * (3 s)^2 = 44.1 m.

Therefore, the ball was thrown from a height of 44.1 meters.

Answer 2

Assume no air resistance, and g = 9.8 m/s².

Let
x = angle that the initial velocity makes with the horizontal.
u = 30 cos(x), horizontal velocity
v = 30 sin(x), vertical launch velocity

The horizontal distance traveled is 55 m, therefore the time of flight is
t = 55/[30 cos(x)] = 1.8333 sec(x) s

With regard to the vertical velocity, and the time of flight,obtain
[30 sin(x)]*(1.8333 sec(x)) + (1/2)*(-9.8)*(1.8333 sec(x))² = 0
55 tan(x) - 16.469 sec²x = 0
55 tan(x) - 16.469[1 + tan²x] = 0
16.469 tan²x - 55 tan(x) + 16.469 = 0
tan²x - 3.3396 tan(x) + 1 = 0

Solve with the quadratic formula.
tan(x) = 0.5[3.3396 +/- √(7.153)] = 3.007 or 0.3326
Therefore
x = 71.6° or x = 18.4°

The time of flight is
t = 1.8333 sec(x) = 5.8096 s or 1.932 s
The initial vertical velocity is
v = 30 sin(x) = 28.467 m/s or 9.468 m/s
The horizontal velocity is
u = 30 cos(x) = 9.467 m/s or 28.469 m/s

If t = 5.8096 s,
u*t = 9.467*5.8096 = 55 m (Correct)
or
u*t = 28.469*15.8096 = 165.4 m (Incorrect)

Therefore, reject x = 18.4°. The correct solution is
t = 5.8096 s
x = 71.6°
u = 9.467 m/s
v = 28.467 m/s

The height from which the ball was thrown is
h = 28.467*5.8096 - 0.5*9.8*5.8096² = -110.4 m
The ball was thrown from a height of 110.4 m

Answer: h = 110.4 m