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.
a) Give the equation of motion of this object.
b) Determine the velocity of object at any time t.
c) Find out the acceleration of object at any time t.
a) To derive the equation of motion for this object, we will use Newton's second law of motion.
The net force acting on the object is given by the difference between the applied force and the air resistance force:
Net force = F - bv.
Since the force is in the horizontal direction, we only consider the horizontal component of the motion. According to Newton's second law, the net force is equal to the mass of the object multiplied by its acceleration:
m * acceleration = F - bv.
Simplifying the equation, we get:
acceleration = (F - bv) / m.
b) To find the velocity of the object at any given time t, we need to integrate the equation of motion with respect to time.
Integrating both sides of the equation, we have:
∫(dv/dt) dt = ∫[(F - bv) / m] dt.
On the left-hand side, the integral of dv/dt with respect to t gives us the velocity v. On the right-hand side, the integral of (F - bv) / m with respect to t gives us:
v = (Ft / m) - (b/m) * ∫v dt.
To evaluate the integral on the right-hand side, we take into account that v is a function of time t. We can rearrange the equation:
v + (b/m) * ∫v dt = (Ft / m).
The integral (b/m) * ∫v dt yields the term (b/m) * vt + C, where C is the constant of integration. Rearranging the equation again, we obtain:
v + (b/m) * vt + (b/m) * C = (Ft / m).
Simplifying, we find:
v (1 + (b/m)t) = (Ft / m) - (b/m) * C.
Finally, solving for v:
v = [(F / m) - (b/m) * C] / (1 + (b/m)t).
c) The acceleration of the object at any given time t is given by the equation of motion:
acceleration = (F - bv) / m.