Motion in a Resistive medium

In mechanics we used to study the motion of the objects without considering the resistive force they are present in out atmosphere. Any measurement we make in real life will not be accurate if we don’t consider the resistive forces in the Motion of an object.

          Hence studying about the motion with resistive forces gives us a clear understanding of what will really happen in nature. We will encounter some mathematics to get the understanding.

         

          It was experimentally understood that the resistive forces are proportional to the quadratic relation with velocity.

First to get a simple understanding we will make it a linear relation with velocity in atmosphere.

Note: All these Forces, velocities and acceleration are vector quantities. Dealing with vector quantities is quite complicated than the scalars. So we always used to split any motion into its components where we can deal it as the scalar quantities.

So we split a three dimensional motion into its fundamental components into two horizontal components and a vertical component. That is in a Right Hand Cartesian Coordinate system, we consider the x and y component as horizontal components since it is coplanar with the ground. And we took the z component as the vertical one as it is perpendicular to the plane of the ground. The ground is considered to be flat because we are taking only a small portion of earth that is a good approximation.  

Thus we are going to analyze the motion in three components but in two ways where the motion in x and y direction is similar.

Part – 1

Motion of the object is coplanar with the plane of the ground. So, the velocity will be considered as either the x or y component of velocity. Both components will give similar results.

When there is no other force in the horizontal direction using Newton’s Second Law,

m dv(x)/dt  (or) m dv(y)/dt = -m β v(x) or –m β v(y)

or simply,   we can write v(x) = v(y) = vx   

Hence it gives us.

                   m dvx  /dt = - m β vx   ,

β is constant that depends on the shape and size of the object and on the resistive medium. The negative sign indicates that the resistive force is in the opposite direction of the motion of the particle.

So,              dvx /dt = - β vx                        →               dvx / vx = - β dt

Integrating dvx  from “Vox” to “Vtx” and time from “0” to “t”

                   ∫ dvx / vx = -β  ∫ dt        

→               Vtx = Vox e^-βt   …..                eq.(1)

         

From this result we can know that,

·       If there is no resistive force that is β = 0 then   Vtx = Vox . The velocity will not change in that direction until there is an extra force other than the resistive force.

·       The velocity in the planar direction to the ground decreases exponentially with time.

  

Part – 2

Now we will talk about the vertical motion where we will some non-intuitive concepts.

Motion of the object is perpendicular to the plane of the ground. So, the velocity will be considered as the z component of velocity that is perpendicular to the ground. 

In the same way as we done before, Using Newton’s second law we can write the equation of motion in perpendicular direction to the plane of ground as follows,

but now we have to include gravitational force. Taking velocity perpendicular to plane as vz

                             m dvz / dt = - mg - β m vz

depending on the direction of the velocity, the resistive force will change its direction.

                            

                             dvz / dt = - g - β vz

                       

→               dvz /( g + β vz ) = - dt

Integrating vz  from “Voz” to “Vtz” and time from “0” to “t” gives

→               Vtz = Voz e^-βt + g/β (e^-βt ­– 1)  …………….                        eq.(2)

Thus we have our second equation.

From the equation (2) we can understand,

·       Velocity in perpendicular direction tends to a limiting value when time goes to infinite value. The limiting value is equal to  “-g/β”. 

·       And this limiting value is called the “The terminal velocity”.

·       From the equation it is known that if the initial velocity is lower than the terminal velocity, the object will tend to increase its speed until it reaches terminal velocity.

·       But if the initial velocity is higher than the terminal velocity the object will start to slow down until it attends the terminal velocity.

·       The terminal velocity of a skydiver in free fall is nearly equal to

200 kilometers per hour.

But to get more accurate results, the resistive force should be treated as quadratic relations with velocity. The mathematics to solve differential equation with quadratic relations is too much complicated than the linear relations. However we will give an attempt to that.