a soccer player kicks a ball into the air at an angle of 36.0 above the horizontal. the initial velocity of the ball is +30.0m/s. how long is the soccer ball in the air?
in the vertical:
hf=hi+viv*t-1/2 g t^2
0=0+30sin36*t-4.9t^2
t(30sin36-4.9t)=0
t=0 and t= 30sin36/4.9
To find how long the soccer ball is in the air, we can break down the vertical and horizontal components of its motion.
Given:
Initial velocity (v0) = 30.0 m/s
Launch angle (θ) = 36.0 degrees
Step 1: Find the vertical component of velocity (v0y):
v0y = v0 * sin(θ)
= 30.0 m/s * sin(36.0°)
≈ 18.2 m/s
Step 2: Find the time it takes for the ball to reach its highest point (t1):
Using the formula for vertical motion:
v = v0y - g * t
0 = 18.2 m/s - 9.8 m/s^2 * t1
t1 = 18.2 m/s / 9.8 m/s^2
≈ 1.86 s
Step 3: Find the total time of flight (t):
Since the ball takes the same amount of time to go up as it does to come back down, the total time of flight is twice t1.
t = 2 * t1
≈ 2 * 1.86 s
≈ 3.72 s
Therefore, the soccer ball is in the air for approximately 3.72 seconds.
To find how long the soccer ball is in the air, we can use the motion equations for projectile motion.
First, let's break down the initial velocity into its horizontal and vertical components:
Vertical component:
The initial velocity of the ball is +30.0 m/s, and it is kicked at an angle of 36.0° above the horizontal. To find the vertical component of the initial velocity (Vy), we can use the equation: Vy = V * sin(angle)
Vy = 30.0 m/s * sin(36.0°)
Vy ≈ 30.0 m/s * 0.5878
Vy ≈ 17.634 m/s (rounded to 3 decimal places)
Horizontal component:
To find the horizontal component of the initial velocity (Vx), we can use the equation: Vx = V * cos(angle)
Vx = 30.0 m/s * cos(36.0°)
Vx ≈ 30.0 m/s * 0.8090
Vx ≈ 24.270 m/s (rounded to 3 decimal places)
Next, we can calculate the time it takes for the ball to reach its maximum height using the equation for vertical motion: Vy = Voy + (-g)t, where Voy is the initial vertical velocity and g is the acceleration due to gravity (approximately 9.8 m/s^2). At the maximum height, the vertical velocity becomes zero.
0 = 17.634 m/s + (-9.8 m/s^2)t
9.8 m/s^2t = 17.634 m/s
t = 17.634 m/s ÷ 9.8 m/s^2
t ≈ 1.8 seconds (rounded to 1 decimal place)
Since the time to reach the maximum height is the same as the time to descend to the ground, we can double the time to find the total time the ball is in the air.
Total time in the air = 2 * t
Total time in the air ≈ 2 * 1.8 seconds
Total time in the air ≈ 3.6 seconds
Therefore, the soccer ball is in the air for approximately 3.6 seconds.