A quarterback (QB) is running at 3.2 m/s at an angle of 25 degrees with x axis as shown in the sketch. At the moment he's at the origin (x = y = 0) he throws the football to a receiver which is stationary located at the coordinates x = 10 meter, y = 6 meter. The total velocity of the ball (accounting for velocity out of the quarterback's hand in addition to the quarterback's velocity) must have a magnitude of 16.2 m/s in order to make it to the receiver in time. With respect to the x axis, what angle must the ball be thrown in degrees? (Hint: this is similar to the football throw problem done in class except the QB is running at an angle and the receiver isn't directly "in front" of the QB. The setup of the problem is the same though - it's just a vector addition/subtraction problem so find all the components that can be found and perform a vector addition/subtraction. Also, remember the total velocity of the QB in addition to the ball must be in the direction from the QB to the receiver such that it will make it to the receiver.)
See previous post: Thu, 9-24-15, 7:24 PM
To find the angle at which the ball must be thrown, we can break down the velocities of the quarterback and the ball into their respective x and y components.
1. Calculate the x and y components of the quarterback's velocity:
- The x-component of the quarterback's velocity can be calculated using the formula: vx = v * cos(θ), where v is the magnitude of the quarterback's velocity and θ is the angle between the velocity vector and the x-axis.
In this case, the magnitude of the quarterback's velocity is 3.2 m/s, and the angle is 25 degrees. Thus, we have: vx = 3.2 * cos(25°).
- The y-component of the quarterback's velocity can be calculated using the formula: vy = v * sin(θ), where v is the magnitude of the quarterback's velocity and θ is the angle between the velocity vector and the x-axis.
In this case, the magnitude of the quarterback's velocity is 3.2 m/s, and the angle is 25 degrees. Thus, we have: vy = 3.2 * sin(25°).
2. Calculate the x and y components of the ball's velocity:
- The x-component of the ball's velocity will be the same as the x-component of the quarterback's velocity, as the ball is thrown in the same direction as the quarterback's velocity.
Therefore, we have: vbx = 3.2 * cos(25°).
- The y-component of the ball's velocity can be determined by considering the total velocity of the ball and the quarterback. Let's denote the y-component of the ball's velocity as vby. The total velocity is 16.2 m/s, and the y-component of the quarterback's velocity is vy = 3.2 * sin(25°).
Since the total velocity must have a magnitude of 16.2 m/s, and the x-components are the same, we can use the Pythagorean theorem to relate the two y-components:
vby^2 + vy^2 = 16.2^2
vby^2 + (3.2 * sin(25°))^2 = 16.2^2
Solve for vby: vby = sqrt(16.2^2 - (3.2 * sin(25°))^2)
3. To find the angle at which the ball must be thrown, we can use the inverse tangent function (arctan) to calculate the angle with respect to the x-axis. Let's denote this angle as θb.
θb = arctan(vby / vbx)
Now, we can substitute the calculated values into the equation to find the angle at which the ball must be thrown.
θb = arctan(sqrt(16.2^2 - (3.2 * sin(25°))^2) / (3.2 * cos(25°)))