Assume acceleration due to gravity=10 m/s. A ball is thrown vertically upward from top of a building with speed of 20 m/s. On the way back ball misses the building and lands on the ground. Total time the ball is in air is 6 seconds. Determine the height of the building with speed of 30 m/s.
The ball's rise time derives from 0 = 20 - 10t or t = 2sec.
During this 2 sec. period, it climbs to
h = 20(2) - 5(2)^2 = 20m.
The time for the ball to reach the ground is 6 - 2 = 4 sec.
The distance covered in 4 sec. derives from
h = 0 + 5(4)2 = 80m. making the building 80 - 20 = 60m high.
To determine the height of the building when the ball is thrown with the speed of 30 m/s, we need to first analyze the motion of the ball.
Let's break the motion into two parts: when the ball is going up and when it's coming down.
1. When the ball is going up:
The initial vertical velocity, u = 30 m/s (upwards)
Acceleration due to gravity, a = -10 m/s² (negative because it acts downwards)
We can use the formula to calculate the time taken for the ball to reach its maximum height during the upward journey:
Final velocity, v = 0 (at maximum height)
Using the equation v = u + at, we can solve for t:
0 = 30 - 10t
10t = 30
t = 3 seconds
So, it took 3 seconds for the ball to reach its maximum height.
To find the maximum height, we can use the kinematic equation:
h = u*t + (1/2)*a*t²
h = 30 * 3 + (1/2) * (-10) * (3)²
h = 90 - 45
h = 45 meters (maximum height)
2. When the ball is coming down:
Now, we know that the total time the ball is in the air is 6 seconds. Since it took 3 seconds for the ball to reach its maximum height, it will take another 3 seconds for the ball to come back down.
From the maximum height down to the ground, the initial vertical velocity, u = 0 (at the top)
Acceleration due to gravity, a = -10 m/s² (still negative because it acts downwards)
Using the equation v = u + at, we can solve for v:
v = u + at
v = 0 + (-10) * 3
v = -30 m/s
The negative sign indicates the direction is downward.
Now, using the formula h = u*t + (1/2)*a*t², we can calculate the distance traveled during the downward journey:
h = 0 * 3 + (1/2) * (-10) * (3)²
h = -45 meters (negative sign indicates the direction is downward)
Therefore, the height of the building can be calculated by summing up the heights from both parts of the motion:
Height of the building = Maximum height + Distance traveled during the downward journey
Height of the building = 45 meters + (-45) meters
Height of the building = 0 meters
So, the height of the building when the ball is thrown with a speed of 30 m/s is 0 meters (ground level).