The statement that the time for an object to go up is equal to the time for it to come down is a simplification often used in introductory physics or in idealized scenarios. It stems from the assumption of symmetry and neglects certain factors that can affect the motion of the object.
In a simplified scenario where only gravitational forces are considered, assuming no air resistance or other external forces, the time it takes for an object to go up to a certain height and then come back down to the same height is indeed equal. This is known as the time of flight symmetry.
To understand why this is the case, we can consider the motion of an object launched vertically upward. Initially, it moves against the force of gravity, gradually slowing down until it reaches its highest point (the peak of its trajectory). From there, it begins to descend, accelerating due to gravity, until it returns to its original height.
The key concept is that the acceleration due to gravity is constant near the Earth's surface, regardless of whether the object is moving up or down. Since the acceleration is the same, and the initial and final heights are the same, the time it takes for the object to reach the peak (go up) is equal to the time it takes to return to the original height (come down). This assumes no loss of energy due to factors like air resistance or friction, which are typically ignored in this simplified scenario.
However, it's important to note that in real-world scenarios, various factors can come into play and cause deviations from this idealized symmetry. Air resistance, differences in initial and final conditions, variations in the object's shape or mass distribution, and other external forces can influence the object's motion and lead to differences in the time it takes for it to go up compared to the time it takes for it to come down.