A quarterback makes a pass, throwing the ball at 21 m/s and an angle of 60° above the horizontal.
A) What is the maximum height of the ball in flight?
B) How long does it take to complete the pass?
C) How many yards (yd) does he manage to throw the ball? (1 yard = 0.9144 meters)
D) The receiver has a 22 yard lead from the quarter back by the time he throws the ball. How fast must he run in order to catch the ball when it lands?
A) h= vₒ²•sin²α/2g,
B) t= 2vₒ•sinα/g
C) L=vₒ²•sin2α/g,
To find the answers to the given questions, we can use the principles of projectile motion.
First, let's break down the problem and identify the given information:
Initial velocity (v) = 21 m/s
Launch angle (θ) = 60°
Acceleration due to gravity (g) = 9.8 m/s²
Times (t) = ?
Distance (d) = ?
A) To find the maximum height of the ball in flight:
The maximum height (H) occurs when the vertical velocity component (Vy) becomes zero.
Using the formula: Vy = v * sin(θ) - g * t
Setting Vy = 0, we can solve for t.
0 = v * sin(θ) - g * t
t = v * sin(θ) / g
Now, plug in the given values:
t = 21 * sin(60°) / 9.8
Using a calculator, we find t ≈ 2.14 seconds.
To find the maximum height (H), we can use the formula:
H = v * sin(θ) * t - 0.5 * g * t²
H = 21 * sin(60°) * 2.14 - 0.5 * 9.8 * (2.14)²
Using a calculator, we find H ≈ 15.1 meters.
Therefore, the maximum height of the ball in flight is approximately 15.1 meters.
B) To find the time it takes to complete the pass:
Since the ball is in flight for the entire pass, we can use twice the time we calculated above:
Total time = 2 * t
Total time ≈ 2 * 2.14
Total time ≈ 4.28 seconds
Therefore, it takes approximately 4.28 seconds to complete the pass.
C) To find how many yards the ball is thrown, we need to find the horizontal distance traveled (d).
The horizontal velocity component (Vx) of the ball remains constant throughout the motion.
Vx = v * cos(θ)
d = Vx * t
Let's calculate Vx:
Vx = 21 * cos(60°)
Using a calculator, we find Vx ≈ 10.5 m/s.
Now, let's calculate the distance (d):
d = 10.5 * 4.28
Using a calculator, we find d ≈ 44.94 meters.
To convert this distance to yards:
d_yards = d * (1 yard / 0.9144 meters)
Using a calculator, we find d_yards ≈ 49.13 yards.
Therefore, the quarterback manages to throw the ball approximately 49.13 yards.
D) To find the speed the receiver must run to catch the ball when it lands:
Since the receiver has a 22-yard lead, we need to find the time it takes for the ball to travel 22 yards.
Using d = Vx * t, we can solve for t:
22 = 10.5 * t
Solving for t, we find t ≈ 2.1 seconds.
Now, we can find the velocity (V) the receiver must run to catch the ball:
V = d / t
V = 22 / 2.1
Using a calculator, we find V ≈ 10.47 yards/second.
Therefore, the receiver must run at approximately 10.47 yards/second to catch the ball when it lands.