12. A ball is kicked with an initial velocity if 40 m/s from the ground at an angle of 30° to the
horizontal. (Use g = 10m/s²) Calculate
a) Horizontal and vertical component of initial velocity
b) The vertical velocity after t = 1s,
c) Position (r = xi+ yj) after t= 1s,
d) total time of flight (time)
e) maximum height
f) Range of projectile
a) To find the horizontal and vertical components of the initial velocity, we can use the equations:
Horizontal component: Vx = V * cos(θ)
Vertical component: Vy = V * sin(θ)
Given:
V = 40 m/s
θ = 30°
Using these values, we can calculate:
Vx = 40 * cos(30°) ≈ 34.64 m/s
Vy = 40 * sin(30°) = 20 m/s
b) To find the vertical velocity after t = 1s, we can use the equation:
Vertical velocity after time t = Vy - g * t
Given:
Vy = 20 m/s
g = 10 m/s²
t = 1s
Using these values, we can calculate:
Vertical velocity after t = 1s = 20 - 10 * 1 = 20 - 10 = 10 m/s
c) To find the position (r = xi + yj) after t = 1s, we can use the equations:
Horizontal position: x = Vx * t
Vertical position: y = Vy * t - (1/2) * g * t²
Given:
Vx = 34.64 m/s
Vy = 20 m/s
g = 10 m/s²
t = 1s
Using these values, we can calculate:
x = 34.64 * 1 = 34.64 m
y = 20 * 1 - (1/2) * 10 * (1)^2 = 20 - 5 = 15 m
Therefore, the position after t = 1s is r = 34.64i + 15j.
d) The total time of flight (time) is the time it takes for the projectile to reach the maximum height and then return to the ground. We can find it using the equation:
Time of flight = (2 * Vy) / g
Given:
Vy = 20 m/s
g = 10 m/s²
Using these values, we can calculate:
Time of flight = (2 * 20) / 10 = 4 s
Therefore, the total time of flight is 4 seconds.
e) To find the maximum height, we can use the equation:
Maximum height = (Vy²) / (2 * g)
Given:
Vy = 20 m/s
g = 10 m/s²
Using these values, we can calculate:
Maximum height = (20^2) / (2 * 10) = 400 / 20 = 20 m
Therefore, the maximum height is 20 meters.
f) To find the range of the projectile, we can use the equation:
Range = Vx * time
Given:
Vx = 34.64 m/s
time = 4 s
Using these values, we can calculate:
Range = 34.64 * 4 = 138.56 m
Therefore, the range of the projectile is 138.56 meters.