Integral problem question (energy and momentum)

[Yura]

i got a sheet full of a lot of questions on integrations and differentiations and got them all except for two of them. in both the questions i run into the same type of problem so if i get one then i can probably work out the other using the same method.
here's the question that's troubling me:

(i've worked out part a) but i can't seem to get out part b). )

a particle with mass 80g is acted on by a force which decreases uniformly with respect to displacement from 10N to zero over 2 metres.

a) calculate the maximum velocity of the particle, given v(0) = 0.

b) find the time for which the force becomes zero.

the part a) answer that i got was 9m/s.
for the part b) its like I am missing some information but seeing as there's two questions like that on the sheet I am guessing there's a method to do this. what i think i need is a function of the Force to x so that i can use
dx/dt = dx/dF * dF/dt.
but unless i have that infomation, right now I am clueless on how to continue.


EDIT: I've done a little more on it but i ended up with the integral of
(-62.5x^2 + 250x)^(-1-2) dx ... but i don't know how to do this integration.

much appreciated if someone can show me how this is done.
thanks.

 

[HallsofIvy]

" what i think i need is a function of the Force to x so that i can use
dx/dt = dx/dF * dF/dt."

But you are given that! F(x) is linear because "F decreases uniformly from 2 N to 0 over 2 meters".
Like any linear function, F can be written as F(x)= ax+ b and you know that
F(0)= 10, F(2)= 0.

 

[M98Ranger]

It seems like you are trying to solve an integral problem involving energy and momentum. It's great that you were able to solve part a) of the problem and find the maximum velocity of the particle. However, in order to solve part b), you need to understand a few concepts related to energy and momentum.

Firstly, you need to understand that the force acting on the particle is related to its acceleration through the equation F=ma, where m is the mass of the particle and a is its acceleration. In this problem, the force is decreasing uniformly from 10N to 0N over a distance of 2 meters. This means that the acceleration of the particle is also changing uniformly over this distance.

Now, to find the time for which the force becomes zero, you need to use the concept of work and energy. The work done by a force is equal to the change in kinetic energy of the particle. In this case, the initial kinetic energy of the particle is zero and the final kinetic energy is given by 1/2 mv^2 (where v is the velocity of the particle). Therefore, using the work-energy theorem, you can set up the following equation:

Work done by the force = Change in kinetic energy
=> Force x Displacement = 1/2 mv^2

Substituting the given values, we get:

10N x 2m = 1/2 (0.08kg) v^2
=> v^2 = 40/0.08 = 500
=> v = √500 = 22.36 m/s

So, the maximum velocity of the particle is 22.36 m/s. Now, to find the time for which the force becomes zero, you can use the equation of motion:

v = u + at

where u is the initial velocity (which is 0 in this case), a is the acceleration (which is changing uniformly from 10N to 0N over a distance of 2 meters), and t is the time for which the force becomes zero.

Therefore, we can write:

22.36 = 0 + (10/0.08)t
=> t = 22.36 x 0.08/10 = 0.1789 seconds

So, the time for which the force becomes zero is approximately 0.1789 seconds.

In order to solve the integration problem, you need to use the concept of work and energy