Sure, I can help you with that!
To find the answer to each question, we can use the equations of motion for projectile motion. These equations are:
1) Vertical displacement, y = ut + (1/2)gt^2
2) Final velocity in the vertical direction, vy = u + gt
3) Final velocity in the horizontal direction, vx = u
(a) To find the speed at which the ball hits the ground, we need to find the magnitude of the final velocity vector. The final velocity vector is the resultant of the horizontal and vertical components of velocity.
Since the ball is kicked horizontally, there is no change in its horizontal velocity. So, the horizontal component of the velocity remains 21 m/s.
In the vertical direction, the initial velocity is 16 m/s, and the acceleration due to gravity is approximately 9.8 m/s^2 downwards. The final velocity in the vertical direction can be found using equation (2):
vy = u + gt
vy = 16 m/s + (9.8 m/s^2)(t)
At the moment the ball hits the ground, the vertical displacement is zero. So, we can find the time it takes for the ball to hit the ground by solving for t in equation (1), with y = 0 and u = 16 m/s:
0 = (16 m/s)t + (1/2)(9.8 m/s^2)(t^2)
0 = 16t + 4.9t^2
Solving this quadratic equation will give us the time it takes for the ball to hit the ground.
Once we have the time, we can substitute it into equation (2) to find the final velocity in the vertical direction, vy. Finally, we can use the Pythagorean theorem to find the speed at which the ball hits the ground by taking the square root of the sum of the squares of the horizontal and vertical components of velocity.
(b) To find how long the ball remains in the air, we will use the same equation as in part (a), y = ut + (1/2)gt^2, but this time we solve for the time it takes for the ball to reach its highest point, where y = maximum height.
(c) To find the maximum height attained by the ball, we can find the vertical displacement at the point where the ball reaches its highest point.
I hope this explanation helps you understand how to find the answers to these questions! Let me know if you need further assistance with the calculations.