A little annoying doubt -- Initial vertical speed of a jumping flea

[Clockclocle]

Homework Statement

If a flea can jump straight up to a height of 0.540 m, what is its initial speed as it leaves the ground?

Relevant Equations

$ \v^2_{x} - \v^2+{0} =2ad $

I know the solution is solved by the equation $ \v^2_{x} - \v^2+{0} =2ad $. But in order to the flea can jump, It must exert a force on the ground at time t=0. Do I have to include this force to substract from earth gravity? Or since this force only appear at time t=0 only while g appear the whole time interval (0,N] for some N then only earth acceleration involved ? What if I assume an infinite weird force join in at countable infinite instance does it affect the calculation? Does this initial force is responsible for initial velocity?

 

[mjc123]

No. The instant it leaves the ground, the only force acting on it is gravity.

 

[Clockclocle]

No. The instant it leaves the ground, the only force acting on it is gravity.

So you only consider the force which appear on the time interval (0,1)? I suppose the flea take 1 second to jump at the height 0.54m? What if at each t=1/n for integer N an upward force of 5N applied on the flea. Do it affect the normal calculation

 

[kuruman]

Do I have to include this force to substract from earth gravity?

You do if you are asked to find the average force that the flea exerts on the ground and you are given the time $\Delta t$ during which the flea exerts this force. Then this average force is $$F_{\text{avg}}=\frac{mv}{\Delta t}-mg=\frac{m\sqrt{2gd}}{\Delta t}-mg.$$

 

[jbriggs444]

So you only consider the force which appear on the time interval (0,1)? I suppose the flea take 1 second to jump at the height 0.54m? What if at each t=1/n for integer N an upward force of 5N applied on the flea. Do it affect the normal calculation

One way to think about this is with calculus. The change in the velocity of the flea is the integral of the acceleration of the flea from the start of an interval to the present time.

If you integrate a function that has little 5N blips for each $t$ that matches $\frac{1}{N}$ for integer n then the integral will match the integral of a function that has no such blips. The blips do not matter.

Another way of thinking about the situation to treat the launch as an impulsive event. That is, an event that takes negligible time. There will be a very large force for a very brief time during which the flea will move by a negligible displacement but attain the requisite initial velocity.

If you integrate a function that has a very large blip at $t=0$ that takes a very short time, the size of the blip matters greatly. Mathematically, one can model such a blip using a Dirac Delta Function.

If one is trying to figure out an average force over an interval that begins or ends with an impulsive event then it is important to be clear on whether the average includes or excludes the event.

For example, the average force from the floor on an ideal basketball starting at one bounce and ending at the next. Is it $mg$, $2mg$ or zero?

 

[gyorgiy]

The "clue" is in the meaning of "as it leaves the ground":
This statement implies two things:
Its starting height is zero, and
Any force the ground exerts on it lasts for no time after that moment: so the effect on speed will be nil.

There are approximations implied in the question which could cause confusion:
Not every part of the flea will be travelling at the same speed at the instance of take-off: for example, the bottom of its leg is still stationary but will rapidly accelerate to the speed of the remainder of the body (and slow down the faster bits in the process).
The flea's leg is extended downwards at the instant of take off. It won't be extended as far when it reaches the top of the trajectory
However, the above are tricial compared with ignoring air resistance, so this is a case of "accept that it's an approximation, and calculate" - at least they have taken the trouble to make the numbers easy,