A quarterback throws a spiral to a receiver that is 40m away. The football has a spin of 14 rev/s and went through 58 revolutions before being caught. if the ball is caught at the same height it was thrown at, at what velocity and at what angle should the quarterback throw the ball?
To determine the velocity and angle at which the quarterback should throw the ball, we can use the concepts of projectile motion and kinematics. Here's how you can calculate it step by step:
Step 1: Find the time taken by the ball to reach the receiver.
The time can be calculated using the formula: Time = Number of revolutions / Spin rate
In this case, the spin rate is given as 14 revolutions per second, and the number of revolutions is 58.
Time = 58 revolutions / 14 rev/s
Time ≈ 4.14 seconds
Step 2: Calculate the initial velocity of the ball.
We can use the formula: Distance = Initial Velocity x Time + 0.5 x Acceleration x Time^2
Since the ball is caught at the same height it was thrown at, the initial vertical velocity is 0. Therefore, we only need to consider the horizontal distance.
Distance = Initial horizontal velocity x Time
40m = Initial horizontal velocity x 4.14s
Initial horizontal velocity ≈ 9.66 m/s
Step 3: Determine the launch angle of the throw.
We can use the formula: Range = (Initial Velocity^2 x sin(2θ)) / g,
where Range is the horizontal distance, Initial Velocity is the magnitude of the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (approximately 9.8 m/s^2).
Solving for θ, we get:
θ = 0.5 x arcsin((Range x g) / Initial Velocity^2)
θ = 0.5 x arcsin((40 x 9.8) / 9.66^2)
θ ≈ 45.61 degrees
Therefore, the quarterback should throw the ball with an initial velocity of approximately 9.66 m/s at an angle of approximately 45.61 degrees.
time in air: 58/14 seconds.
distance=vi*cosTheta*t
40=vi*cosTheta*58/14
time in air:
hf=hi+vi*sinTheta*t-1/2 g t^2
0=0+vi*sinTheta*58/14-4.9(58/14)^2
so now, two equations. solve for v in the upperone.
vi=40*14/58*1/cosTheta
put that in the second equation
40*14/58*tanTheta*58/14 -4.9(58/14)^2
solve for tan theta, then theta. Now with theta, you can solve vi.