Question #23642

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

Expert's answer

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

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

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