From the edge of the rooftop of a building, a boy throws a stone at an angle 20.0° above the horizontal. The stone hits the ground 4.60 s later, 104 m away from the base of the building. (Ignore air resistance.)
What is the maximum height H reached by the stone?
(I found that the velocity is 24.06 and the initial height is 65.834, but I cant get the maximum height right.)
horizontal velocity (Vh) is the velocity (V) times the cosine of 20.0º
vertical velocity (Vv) is V times the sine of 20.0º
or __ Vv = Vh * tan(20.0º) = (104/4.60) * tan(20.0º) = 8.23
when the stone hits the ground, H = 0
0 = (-.5 * g * 4.60^2) + (8.23 * 4.60) + Ho
65.826 = Ho
use the axis of symmetry equation to find t at max H (x = -b / 2a)
t max = -8.23 / -9.8 = 0.84
substitute the t back to find H max
Im sorry but what is the equation to use t in? Im confused by your wording.
To determine the maximum height reached by the stone, we need to use the equations of motion. Let's break down the problem into horizontal and vertical components.
1. Vertical Component:
The initial vertical velocity can be calculated using trigonometry. Given the angle of 20.0° above the horizontal, we can find the initial vertical velocity (v_y) using the equation:
v_y = v * sin(theta)
Where v represents the initial velocity (24.06 m/s) and theta represents the angle (20.0°). Plugging in the values:
v_y = 24.06 m/s * sin(20.0°) = 8.14 m/s (rounded to two decimal places)
The time to reach the maximum height is half the total time of flight (t), since at the maximum height, the vertical velocity becomes zero.
t/2 = 4.60 s / 2 = 2.30 s
Using the equation of motion for vertical displacement (y):
y = v_y * t + (1/2) * g * t^2
Where g represents the acceleration due to gravity (9.8 m/s²). Plugging in the values:
y = 8.14 m/s * 2.30 s + (1/2) * 9.8 m/s² * (2.30 s)^2
y = 18.72 m + 25.05 m
y ≈ 43.77 m
So, the maximum height reached by the stone is approximately 43.77 meters.
Note: It seems there may be an error in the initial height calculation you mentioned. The initial height should not be 65.834 m, but rather zero since the stone was thrown from the edge of the rooftop.