A particle is pushed horizontally by a constant horizontal force of magnitude F, which
starts from the rest at x=0 in the positive ݔ direction. During the movement, it is also
acted by the air resistance force that equivalent to bv, where b is the positive coefficient
and v is the instantaneous velocity of the body.
Find out the acceleration of object at any time t.
F - bv = m a
a = (F-bv)/m
that means
dv/dt = F/m - (b/m)v
or
(b/m) v + dv/dt = F/m
let v = c (1-e^-kt)
then dv/dt = cke^-kt
and
(b/m)c -(b/m)ce^-kt +cke^-kt = F/m
when t --> oo, F = bc so c = F/b
when t = 0, c k = F/m
so Fk/b =F/m
so k = b/m
now
v = (F/b)( 1-e^-(bt/m) )
a = dv/dt = (F/b)(b/m)e^-(bt/m)
or
a = (F/m)e^-(bt/m)
To find the acceleration of the object at any time t, we need to consider the forces acting on the particle and apply Newton's second law of motion.
Newton's second law states that the acceleration of an object is equal to the net force acting on it divided by its mass. In this case, the net force is the difference between the applied force and the air resistance force.
Let's break down the problem step by step:
Step 1: Determine the net force acting on the particle.
The net force can be calculated as the difference between the applied force and the air resistance force. Since the applied force is in the horizontal direction, its horizontal component is F. The air resistance force is given by bv, where b is the coefficient and v is the velocity of the particle.
Net force = Applied force - Air resistance force
= F - bv
Step 2: Apply Newton's second law.
Newton's second law states that the acceleration of an object is equal to the net force acting on it divided by its mass. In this case, since the mass of the object has not been given, we can assume it to be m.
Acceleration = Net force / Mass
= (F - bv) / m
Therefore, the acceleration of the object at any time t is given by (F - bv) / m.