Question

A point particle with mass M moves along a straight line under the influence of a net force
that is a function of time, given by:
F_x=Bt²,
where B is positive and constant.
a) If the particle starts from rest, find how long it will take it to cover a distance L.
b) Find the speed of the particle after it has traveled a distance L.
c) Find the work done on the particle by the net force over the distance L.
Write your result in terms of M, B, and L. Check the units/dimensions of each answer.

Answer 1

The net force as a function of time is given by
F = B t²
The acceleration of the mass is
a = F / m = B t² / m
Where m is the mass of the particle
a = dv /d t = B t² / m


At t= 0, v = 0
0 = constant
v = B t³/ 3m
v = dx /dt
dx / dt = B t³/ 3m
dx = ( B t³ / 3 m ) dt
We need to find the time taken to cover the distance L


The time taken to cover the distance L is
t = (12 m L/ B)^1/4
b) The velocity to cover the distance L is,
v = L / t
v = B t⁴ /12m / t
v = B t³ / 12 m
The work done to cover the distance is
W = F . d
Where d is the distance covered
The work done to cover the distance L is
W = F . L = B t² x L = B t² ​​​​​​​L