Answer to Question #23642 in Mechanics | Relativity for rosko

Question

If a football player kicks a football so that it spends 3.0 seconds in the air and travels 50.0 m, then find the velocity and angle of the kick. (22m/s, 41 degrees) , I need step by step solutions. please

Answer 1

Let vx(t) and vy(t) be the horizontal and verticalcoordinates of velocity of the ball at time t.

There is a constant gravitation force
F = mg
acting on the ball during its fly.
This force is directed to the ground and is opposite tovy.

Hence the ball moves uniformly along x-axis and withconstant acceleration -g along y-axis, so vx is constant:
vx(t) = vx(0)
and
vy(t) = vy(0) -g*t.
Thus the ball moved t=3 seconds and fly s=50.0 m, whenceits horizontal velocity, i.e. vx(t) is equal to
vx = s/t = 50/3= 16.667 m/s

Further notice that the ball flies up the same time as itgoes down,
and at the maximal point the time was 1.5 seconds and itsvelocity vy=0, whence
vy(1.5) = vy(0)- 9.8 * 1.5 = 0
so,
vy(0) = 9.8 *1.5 = 14.7 m/s.
Hence the full initial velocity is v = sqrt(vx(0)^2+vy(0)^2)
=sqrt(16.667^2+14.7^2)
=sqrt(277.79+216.09)
= sqrt(493.88)
= 22.223 m/s

Let alpha be the angle of the kick.
Then
tan(alpha) =vy/vx = 14.7/16.667 = 0.88198
whence
alpha =acrtan(0.88198)= 0.72277 rad = 41.412 degree