A ball thrown vertically upward and return in20 seconds.find the velocity and the maximum height with explanation

To find the velocity and maximum height of a ball thrown vertically upward and returning in 20 seconds, we can use the equations of motion for vertical motion.

1. Firstly, we need to identify the relevant variables:
- Initial velocity (u): The velocity at the start of the motion.
- Final velocity (v): The velocity at the end of the motion.
- Acceleration (a): The rate at which the velocity changes.
- Time (t): The total time of motion.
- Displacement (s): The change in position.

2. In this case, the ball is thrown vertically upward, which means the initial velocity will be positive and the acceleration due to gravity will act downwards, so it will be negative.

3. We can use the following equations of motion to solve the problem:

Equation 1: v = u + at
Equation 2: s = ut + (1/2)at^2
Equation 3: v^2 = u^2 + 2as

4. Since the ball returns to its initial position, the displacement will be zero (s = 0).

5. Let's start by finding the initial velocity (u). Since the ball returns to its starting point, the final velocity (v) would be the negative of the initial velocity (-u).

Equation 1: v = u + at
-u = u + (-9.8 * 20) [Using -9.8 m/s^2 as the acceleration due to gravity]
-2u = -196
u = 98 m/s

Therefore, the initial velocity (u) of the ball is 98 m/s.

6. Now, let's find the maximum height reached by the ball. At the maximum height, the final velocity will be zero (v = 0).

Equation 1: v = u + at
0 = 98 + (-9.8 * t)
-9.8t = -98
t = 10 s

The ball takes 10 seconds to reach the maximum height.

Equation 2: s = ut + (1/2)at^2
s = 98 * 10 + (1/2) * (-9.8) * (10^2)
s = 980 - 490
s = 490 m

Therefore, the maximum height reached by the ball is 490 meters.

7. Lastly, let's find the final velocity of the ball when it returns in 20 seconds.

Equation 1: v = u + at
v = 98 + (-9.8 * 20)
v = 98 - 196
v = -98 m/s

Therefore, the final velocity of the ball when it returns is -98 m/s (negative indicating the downward direction).

To summarize:
- The initial velocity (u) of the ball is 98 m/s.
- The maximum height reached by the ball is 490 meters.
- The final velocity of the ball when it returns is -98 m/s.