# Velocity problem

Consider a body thrown up with some arbitrary initial velocity. It can be proven that the speed of the body one second before it reaches its maximum height is always the same and is independent of the initial velocity of the body.

How, or rather why is this so? What can be a simple and logical explanation for this? As a special case, it is obvious that the above is not true for any arbitrary instant of the motion. When does it 'start' becoming true, ie is it valid for 2,3... seconds before the body reaches its maximum height? Please excuse me for any wrong concepts in my question. I'm new to physics and am really interested!

• Hint : Think about what happens when it reaches maximum height. – StephenG Apr 14 '18 at 12:49
• The velocity would become zero. – Rishikesh Sarma Apr 14 '18 at 12:51
• And how can you use that to predict the motion before and after that ? – StephenG Apr 14 '18 at 13:11
• I understand that since the acceleration (due to gravity) for both the bodies is same, hence the velocity one second before it reaches the max height must be the same. Right? – Rishikesh Sarma Apr 14 '18 at 13:20
• But how long is this true? – Rishikesh Sarma Apr 14 '18 at 13:21

Work backwards from the object at maximum height when its vertical velocity is zero.

Its speed at time $t$ before or after reaching that height is $gt$ where $g$ is the gravitational field strength.
The only difference will be that the velocity before reaching maximum height will be upwards and the velocity will be downwards after reaching maximum height.

The speed at a time one second before reaching maximum height is thus only dependent on $g$.

As many comments pointed out, this is true because the acceleration is the same. You say: "it is obvious that the above is not true for any arbitrary instant of the motion".

No, it is true, but, if the initial velocity is different, it will take a different number of seconds to reach the top. So, it is true, until one of the two objects you are comparing, going back in time, reaches the time it was launched up in the air. You cannot go back further, but you can for the second object, that was launched at higher speed.

As an aside, if one object was launched at such low speed that it takes less than a second to reach maximum height, that statement is wrong. The authors take one second as a small enough period of time. It could be 1 minute, but then you must assume both objects were launched at initial speeds so high that they take at least one minute to reach maximum height.

• The reason I was getting the OP to do the thinking with a little nudge in the right direction was to help them learn and to avoid breaking our homework-type policy. In general we do not directly answer homework-type questions that don't show effort. Please keep this in mind when responding to questions in future. – StephenG Apr 14 '18 at 14:02
• @StephenG This does not look like a homework question to me, rather an explanation that the OP did not quite grasp. The questions: "when does it stop being true, at 2-3 secs, etc" seemed an attempt to understand the explanation, rather than believe blindly the book and memorise concepts. I think he/she deserved an answer. – user Apr 14 '18 at 14:14
• Note that it makes no difference if it takes less than one second to reach the top. The statement is true if it takes a micro second to reach the top. Or any interval at all (assuming the usual conditions, no atmosphere, etc) – garyp Apr 14 '18 at 15:19
• @garyp The statement is that the two objects will have same velocity one second before reaching the top regardless of the initial velocity. If one object is thrown at such low speed that it takes less than a sec to reach max height, it will still be on the ground and the statement is incorrect. – user Apr 14 '18 at 16:07
• I see. You might consider editing to avoid misinterpretation. For example, change the last phrase to "take at least one second to reach maximum height". – garyp Apr 14 '18 at 16:23