Question

F is a position dependent force given by F(x) = -e^-x. Sketch the graphs showing F(t), v(t), and x(t) for initial velocity of 10m/s, initial position of 100m, and mass = 1kg. Mention all salient points.

Answer 1

Newton's equation of motion:

Exponential can be expanded as:

Therefore,

We will approximate to the 1st order term to keep the DE linear:

for small x

since m=1kg,

The complementary function:

The particular Integral:

Therefore the general solution:

Using Boundary condition, x(0)=100

.........(1)

Using boundary condition

...........(2)

Adding equation (1) and (2)

Therefore,

Therefore the position is given by:

x(t):

v(t):

F(t):

At t=0, x=100 m, v=10 m/sec, F=99 N.