A quarterback is asked to throw a football to a receiver that is 32.6 m away. What is the minimum speed that the football must have when it leaves the quarterback's hand? Ignore air resistance. Assume the ball is caught at the same height as it is thrown.
The formula you need to use for the distance R of the pass, which is easily derived, is
R = 32.6 = (Vo^2/g)*sin(2A)
Vo is the velocity at which the ball is thrown
For the minimum required speed, make A = 45 degrees, so that 2A = 90 degrees.
32.6 = Vo^2/g
Vo = 17.9 m/s
L=vₒ²•sin2α/g,
vₒ =sqrt(L•g/sin2α)
min vₒ at sin2α = 1
vₒ =sqrt(L•g) = sqrt(32.6•9.8) =17.87 m/s
To determine the minimum speed the football must have when it leaves the quarterback's hand, we can use the principles of projectile motion. In this scenario, the football is being thrown horizontally, and we need to calculate the initial speed required to reach the receiver 32.6 m away.
Let's break down the motion of the football:
1. Horizontal motion: Since the football is thrown horizontally, there is no acceleration in the horizontal direction. Therefore, the velocity in the horizontal direction remains constant throughout the motion.
2. Vertical motion: The football will experience vertical acceleration due to gravity. However, since we assume the ball is caught at the same height it is thrown, the vertical displacement is zero. As a result, the time of flight and the vertical velocity at the catching point are not considered in this calculation.
Now, we can calculate the minimum speed required by following these steps:
Step 1: Determine the time it takes for the football to travel horizontally.
The formula for calculating the time of flight in the horizontal direction is:
time = distance / velocity_horizontal
Since the velocity in the horizontal direction remains constant, we can write:
time = distance / speed
Substituting the given distance of 32.6 m:
time = 32.6 m / speed
Step 2: Calculate the minimum initial speed required.
To calculate the minimum initial speed, we need to determine the time it takes for the football to travel horizontally and the distance it travels in that time.
Since the vertical displacement is zero, we do not need to consider the vertical component of motion. Thus, the horizontal distance traveled is the same as the distance between the quarterback and the receiver.
Using the formula: distance = speed × time
We can rearrange the formula to solve for speed:
speed = distance / time
Substituting the given distance and time:
speed = 32.6 m / (32.6 m / speed)
Simplifying the equation:
speed^2 = 32.6 m^2 / (32.6 m / speed)
Multiplying both sides of the equation by (32.6 m / speed):
speed^3 = 32.6 m^2
Taking the cube root of both sides:
speed = ∛(32.6 m^2)
Evaluating the cube root:
speed ≈ ∛(32.6) m/s
Finally, using a calculator, we find that the minimum speed required for the football to reach the receiver 32.6 m away is approximately 3.072 m/s.