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

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21 March 2020
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 х=0, please find:
* how does velocity and acceleration of the particle depend on time / (time dependence of velocity and acceleration of the particle);
v=dXdt=ax…………1
From equation 1, we have
dXx=adt……2
Integrating both sides of equation 2 to get the equation of position in terms of time (t), we get
dXx=adt….3
2x=at+C…4
When t=0, x=0 from the boundary conditions
Substituting in equation 4,
C=0
Hence
x=at2……5
Squaring both sides,
x=(at)24 m
Velocity in terms of t
v=dXdt=a2t2 m/s
Acceleration in terms of t
a=dVdt=ddta2t2=a22 ms-2
* The average velocity of the particle during the time it runs the first S meters along its path.
To run s=meters,
S=x=(at)24……i