# Physics and Natural Sciences. Engineering Math Problem (Math Problem Sample)

calculation of velocity and boyle's law

source..Subject: Physics and Natural Sciences

1) The particle moves along the positive X-axis, hence its velocity varies as V=αx, where α is a positive constant. Assuming that at the moment t=0 the particle was located at the point x=0, please find:

* how does velocity and acceleration of the particle depend on time / (time dependence of velocity and acceleration of the particle);

Solution:

We express velocity, V, as function of x and t as follows:

V=dxdt=αx

We rearrange the equation to give

dxx=αdt

Integrating

x-1/2 +1-1/2+1=αt+c

2x=αt+c

Where,

c= Integration constant

And therefore

x=αt+c22

Acceleration of the particle is given by

a=d2xdt2=ddtddtαt+c22

d2xdt2=ddtα2αt+c

d2xdt2=α22 m/s2

Taking boundary conditions:

At t=0, x=0

We have

x=αt+c22

0=α*0+c22

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