A quarterback throws the football to a stationary receiver who is 16 m down the ﬁeld.The football is thrown at an initial angle of38.5◦to the ground.The acceleration of gravity is 9.81 m/s2.a) At what initial speed must the quarterback throw the ball for it to reach the receiver?Answer in units of m/s

Question

A quarterback throws the football to a stationary receiver who is 16 m down the ﬁeld.The football is thrown at an initial angle of38.5◦to the ground.The acceleration of gravity is 9.81 m/s2.a) At what initial speed must the quarterback throw the ball for it to reach the receiver?Answer in units of m/s

Answer 1

Well, to give you a straight answer, the initial speed at which the quarterback will need to throw the ball for it to reach the receiver is approximately 18.73 m/s. However, I must say, it's quite the toss! I hope the receiver has good catching skills and doesn't get too tired running that distance!

Answer 2

To find the initial speed with which the quarterback must throw the ball, we can use the kinematic equations of motion.

First, we need to break down the initial velocity of the ball into its horizontal and vertical components. The horizontal component will remain constant throughout the ball's flight, while the vertical component will be affected by gravity.

The horizontal component of the initial velocity (Vx) can be found using the equation:
Vx = Initial velocity * cos(θ)

Here, θ is the angle at which the ball is thrown (38.5°), and cos(θ) represents the cosine function of that angle.

The vertical component of the initial velocity (Vy) can be found using the equation:
Vy = Initial velocity * sin(θ)

Next, let's consider the time it takes for the ball to reach the receiver. The time taken in seconds (t) can be calculated using the equation:
t = [2 * Vy] / g

Here, g represents the acceleration due to gravity (9.81 m/s^2).

Since the ball travels a horizontal distance of 16 m, we have:
16 = Vx * t

Now, we can substitute the values of Vx and t with their respective formulas:
16 = (Initial velocity * cos(θ)) * ([2 * (Initial velocity * sin(θ)) / g])

Simplifying this equation will allow us to solve for the initial velocity:
16 = [2 * Initial velocity^2 * sin(θ) * cos(θ)] / g

Multiplying both sides of the equation by g and rearranging the terms, we get:
2 * Initial velocity^2 * sin(θ) * cos(θ) = 16 * g

Dividing both sides by 2 * sin(θ) * cos(θ), we obtain:
Initial velocity^2 = (16 * g) / (2 * sin(θ) * cos(θ))

Taking the square root of both sides, we find:
Initial velocity = √[(16 * g) / (2 * sin(θ) * cos(θ))]

Now, we can substitute the values of g (9.81 m/s^2) and θ (38.5°) into the formula:
Initial velocity = √[(16 * 9.81) / (2 * sin(38.5) * cos(38.5))]

Evaluating this expression will give us the answer.