a quaterback throws a ball to a stationary reciver wo is 15.2m down the feild. the football is thrown at an initial angle of 42 dagrees. acceleration is 9.81m/s^2. what is the initial speed that quaterback must throw the ball to hit the reciver?
To find the initial speed that the quarterback must throw the ball to hit the receiver, we can use the principles of projectile motion.
First, let's break down the given information:
- The distance between the quarterback and the receiver is 15.2m.
- The initial angle at which the ball is thrown is 42 degrees.
- The acceleration due to gravity is 9.81 m/s^2.
Now, let's analyze the motion of the ball:
1. Resolve the initial velocity into horizontal and vertical components:
The initial velocity can be decomposed into two components:
- The horizontal component remains constant throughout the motion, as there is no acceleration in the horizontal direction.
- The vertical component will change due to acceleration.
2. Calculate the time of flight:
The time taken by the ball to reach the receiver can be determined using the vertical component of the motion. We can use the equation:
h = u * sin(θ) * t - (1/2) * g * t^2
where h = 15.2m (the vertical distance), u is the initial velocity, θ is the angle, g is the acceleration due to gravity, and t is the time of flight.
3. Solve for the initial velocity:
The horizontal distance traveled can be determined using the horizontal component of the motion. We can use the equation:
R = u * cos(θ) * t
where R is the horizontal distance (which is equal to 15.2m), u is the initial velocity, θ is the angle, and t is the time of flight.
Using these equations, let's calculate the initial speed:
Step 1: Calculate the time of flight (t):
h = u * sin(θ) * t - (1/2) * g * t^2
15.2 = u * sin(42°) * t - (1/2) * 9.81 * t^2
Step 2: Solve for the time of flight (t):
Rearrange the equation to solve for t using the quadratic formula or other methods. Once you find the value of t, proceed to the next step.
Step 3: Calculate the initial velocity (u):
R = u * cos(θ) * t
15.2 = u * cos(42°) * t
Rearrange the equation to solve for u:
u = 15.2 / (cos(42°) * t)
By substituting the calculated value of t into the equation, you can find the initial speed (u) required for the quarterback to throw the ball and hit the receiver.