The soccer goal is 23.05 m in front of a soccer player. She kicks the ball giving it a speed of 17.97 m/s at an angle of 25.83 degrees from the horizontal. If the goalie is standing exactly in front of the net, find the speed of the ball just as it reaches the goalie.
Do I need to use delta x = Vx x t to solve for time first? Then what?
I don't know the point of this problem: should air friction be ignored or not?
I think I would ignore air friction, which in reality is silly, but anyway.
Using the intial velocity and angle, solve for the initial vertical and horizontal velocities.
Using the vertical velocity, one needs to find the time of flight.
yf=yi+vyi*t - 4.9 t^2
1=0+7.83t-4.9t^2
4.9t^2-7.83+1=0 using the quadratic formula...
t= (7.83 +- sqrt (61.3-20)/9.8=1/45 sec
Now, where is the ball at that time?
d= 17.97cos25.83 * time=23.45 meters, within the goalie reach of .4 meters.
Now, what is the difference in energy?
Intial energy = 1/2 mv^2
final energy= 1/2 mvf^2 + mgh where h is one meter.
vf^2= vi^2 - 2gh
vf= sqrt (17.97^2 - 2*g*1)=17.42m/s
This problem is subject to a lot of differing assumptions, the main one is friction due to air. This assumes the ball is caught by moving the arms of the goalie.
To solve this problem, you can use the equations of motion for projectile motion. In this case, you can start by finding the initial horizontal and vertical velocities of the ball.
The initial velocity of the ball can be resolved into horizontal and vertical components as follows:
Vx = V * cos(theta)
Vy = V * sin(theta)
Where V is the magnitude of the initial velocity (17.97 m/s in this case), and theta is the angle of the velocity vector from the horizontal (25.83 degrees in this case).
Now, you need to find the time it takes for the ball to reach the goal. To calculate this, you can use the vertical displacement of the ball, assuming that the initial and final vertical positions are the same. Since the ball is kicked on the ground and reaches the ground at the goal, the vertical displacement (Δy) is zero.
To find the time of flight (t), you can use the equation:
Δy = Vy * t + (1/2) * g * t^2
Since Δy is zero, the equation can be simplified to:
0 = Vy * t - (1/2) * g * t^2
Where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Now you have a quadratic equation. You can solve it to find the time t. Once you find the time, you can substitute it back into the equation Vx = Δx / t to find the horizontal velocity component (Vx).
Finally, you can use the Pythagorean theorem to find the magnitude of the velocity just as the ball reaches the goalie:
V_goalie = sqrt(Vx^2 + Vy^2)
By substituting the values you have found into the equation, you can calculate the speed (magnitude) of the ball just as it reaches the goalie.