Question

A player kicks a soccer ball from ground level and sends it flying at an angle of 30 degrees at a speed of 26 m/s. What is the maximum height attained by the ball? Round the answer to the nearest tenth of a meter

Answer 1

8.6 m

Step-by-step explanation:

The motion of a soccer ball is a motion of a projectile, with a uniform motion along the horizontal (x-) direction and an accelerated motion along the vertical (y-) direction, with constant acceleration
`a=g=-9.8 m/s^2` towards the ground (we take upward as positive direction, so acceleration is negative).

The initial velocity along the vertical direction is

`v_(y0) = v_0 sin \theta = (26 m/s)(sin 30^(\circ))=13 m/s`

Now we can consider the motion along the vertical direction only. the vertical velocity at time t is given by:

`v_y(t)=v_(y0) +at`

At the point of maximum height,
`v_y(t)=0`, so we can find the time t at which the ball reaches the maximum height:

`0=v_(y0)+at\\t=-(v_(y0))/(a)=-(13 m/s)/(-9.8 m/s^2)=1.33 s`

And now we can use the equation of motion along the y-axis to find the vertical position of the ball at t=1.33 s, which corresponds to the maximum height of the ball:

`y(t)=v_(y0)t + (1)/(2)at^2=(13 m/s)(1.33 s)+(1)/(2)(-9.8 m/s^2)(1.33 s)^2=8.6 m`