a quaterback throws the football to a staionary receiver who is 23.5 m down the filed. if the football is thrown at an initial angle of 35 to the ground at what inital speed must the quaterback the ball for it to reach the receiver 5s later?
You have over defined this problem.
Once you say 23.5 meters in 5 seconds you have set the horizontal speed
u = 23.5/5
Now if you set the angle at 35 then
u = initial speed * cos 35
so solve for the initial speed
However now we have a dilemna.
Vi = initial speed up = initial speed * sin 35
that means we can calculate how high it will be after 5 seconds
h = height of quarterback hand + Vi (5) - 4.9 (25)
that h better be around the height of a receiver we hope.
To find the initial speed at which the quarterback must throw the football, we can use the following steps:
Step 1: Analyze the components of the projectile motion. The motion of the football can be divided into horizontal (x) and vertical (y) components. In this case, the horizontal distance is given (23.5 m), the time of flight is given (5 s), and the initial angle of projection is given (35°).
Step 2: Determine the initial vertical velocity (Vy0). Since the football is thrown at an angle to the ground, we can use the equation for vertical motion:
Sy = Vy0 * t + (1/2) * g * t^2
In this equation, Sy represents the vertical displacement (23.5 m), t is the time of flight (5 s), and g is the acceleration due to gravity (approximately 9.8 m/s^2). Vy0 refers to the initial vertical velocity, which is what we want to find.
Step 3: Solve the equation for Vy0. Rearranging the equation, we have:
23.5 = Vy0 * 5 + (1/2) * 9.8 * (5)^2
23.5 = 5Vy0 + 122.5
Simplifying the equation:
5Vy0 = -99
Vy0 = -19.8 m/s
Note: The negative sign indicates that the initial vertical velocity is directed downward. This is because the coordinate system's positive direction is upward.
Step 4: Determine the initial horizontal velocity (Vx0). The horizontal velocity is constant throughout the motion, so we don't need any special calculations for it. We can use the equation:
Vx0 = Dx / t
In this equation, Vx0 represents the initial horizontal velocity, Dx is the horizontal distance (23.5 m), and t is the time of flight (5 s).
Calculating Vx0:
Vx0 = 23.5 / 5
Vx0 = 4.7 m/s
Step 5: Calculate the initial speed (V0). The initial speed is the magnitude of the resultant velocity, combining the vertical and horizontal components. We can use the Pythagorean theorem:
V0 = √(Vx0^2 + Vy0^2)
Substituting the values:
V0 = √((4.7)^2 + (-19.8)^2)
V0 = √(22.09 + 392.04)
V0 = √414.13
V0 ≈ 20.35 m/s
Therefore, the quarterback must throw the ball with an initial speed of approximately 20.35 m/s.