To calculate the time it takes for a thrown object to reach half its maximum height, we can use the equations of motion for vertical motion.
Let's assume the object is thrown vertically upwards, reaches a maximum height of 6 meters, and then comes back down. We'll break down the calculation into two parts: ascending and descending.
- Ascending motion: The initial velocity of the object when thrown vertically upwards is denoted as "u" and is equal to the final velocity when the object reaches its maximum height. At the highest point, the velocity becomes zero. The acceleration due to gravity is denoted as "g" and is equal to approximately 9.8 m/s² (assuming no air resistance).
Using the equation of motion:
v = u + at,
where: v = final velocity (0 m/s at the highest point) u = initial velocity a = acceleration (-9.8 m/s²) t = time
0 = u - 9.8t,
Simplifying, we find:
u = 9.8t.
To find the time it takes for the object to reach half its maximum height during the ascending motion, we need to solve for t when the object is at a height of 3 meters (half of 6 meters). We can use another equation of motion:
s = ut + (1/2)at²,
where: s = displacement (3 meters) u = initial velocity a = acceleration (-9.8 m/s²) t = time
3 = (9.8t)t + (1/2)(-9.8)(t²), 3 = 4.9t² - 4.9t², 3 = 0.
Since the equation yields 0, it means the object reaches half its maximum height during the ascending motion at t = 0. This implies that at the beginning of the motion, the object is already at half its maximum height.
- Descending motion: During the descending motion, the object starts at its maximum height and falls back down to the ground. We'll calculate the time it takes for the object to descend from the maximum height of 6 meters to a height of 3 meters.
Using the equation of motion:
s = ut + (1/2)at²,
where: s = displacement (3 meters) u = initial velocity (0 m/s at the highest point) a = acceleration (9.8 m/s²) t = time
3 = (1/2)(9.8)t², 3 = 4.9t², t² = 3/4.9, t ≈ 0.784 seconds.
Therefore, it takes approximately 0.784 seconds for the object to descend from its maximum height of 6 meters to a height of 3 meters.