Find the velocity x and the position x as functions of the time t for a particle

Question

Find the velocity x and the position x as functions of the time t for a particle of mass m, which starts rect at x = 0 and t = 0 subject to the following force functions: F_a = F_0 + ct F_a = F_0 sin ct F_a = F_ 0e^et where F_0 and C are positive constants. Find the velocity x as a function of the displacement x for a particle of mass m, which starts from rest at x = 0, subject to the following force functions: F_x = F_0 + cx F_x = F_0e^-ex F_x = F_0 cos cx where F0 and c an:positive constants. middot 4

Explanation / Answer

force=mass X accelaration=massX changeinvelocity/time

in all the cases the force is not constant.so ordinary equations to findout velocity and displacement wont work.

so

mX dv/dt= Fo+ct

dv/dt=Fo/m +ct/m

dv=(Fo/m)dt+(c/m)tdt

integrating

v=(Fo/m )t+(c/m)t2/2+ constant of integration

here= v=dx/dt

now dx/dt =(Fo/m )t+(c/m)t2/2+ constant of integration

again integrating we will get the values for dx which is displacement.

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