A ball is launched at a velocity of 20 m/s in a direction making an angle of 25° upward with the horizontal. i) calculate maximum height reached by the object ii) Calculate the total flight time of the object iii) Calculate the horizontal range (maximum x above ground) of the object
horizontal
U = 20 cos 25 for the whole trip
so
range = 20 cos 25 * time in air
vertical
initial speed up = Vi = 20 sin 25
v = Vi - g t where t is time up
so
t = Vi/g = (20/9.8) sin 25
time down = t = time up
so
time in air = 2 t
so
range = 2 t * 20 cos 25
for max height, height at t
h = Vi t - 4.9 t^2
To solve this problem, we can break down the initial velocity into its horizontal and vertical components.
Given:
Initial velocity (v): 20 m/s
Launch angle (θ): 25°
i) To calculate the maximum height reached by the object:
Step 1: Determine the vertical component of the initial velocity.
v_vertical = v * sin(θ)
v_vertical = 20 * sin(25°) ≈ 8.598 m/s
Step 2: Use the kinematic equation to calculate the maximum height (h_max).
v_vertical^2 = v_initial^2 - 2*g*h_max
0 = (8.598)^2 - 2*9.8*h_max
h_max = (8.598^2) / (2*9.8) ≈ 3.722 meters
Therefore, the maximum height reached by the object is approximately 3.722 meters.
ii) To calculate the total flight time of the object:
Step 1: Determine the vertical component of time (t_vertical).
t_vertical = (v_vertical) / g
t_vertical = 8.598 / 9.8 ≈ 0.876 seconds
Step 2: Use the time in the vertical direction to calculate the total flight time (t_total).
t_total = 2 * t_vertical
t_total = 2 * 0.876 ≈ 1.752 seconds
Therefore, the total flight time of the object is approximately 1.752 seconds.
iii) To calculate the horizontal range (maximum x above the ground) of the object:
Step 1: Determine the horizontal component of the initial velocity.
v_horizontal = v * cos(θ)
v_horizontal = 20 * cos(25°) ≈ 18.008 m/s
Step 2: Use the horizontal component of velocity and the total flight time to calculate the horizontal range (R).
R = v_horizontal * t_total
R = 18.008 * 1.752 ≈ 31.614 meters
Therefore, the horizontal range (maximum x above the ground) of the object is approximately 31.614 meters.