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

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