Variables of Motion

Variables of Motion

In order to describe the general state of the motion of a physical system, we use the equations of motion. Generally, the equation of motion is governed by the four main variables of position, time, velocity, and acceleration.

For example, in the case of motion of a simple pendulum, the equation of motion of a simple pendulum is given by,

(d2θ/dt2)+ (g/L)θ = 0

Here θ represents position, t represents time, and g represents acceleration.

Now let’s take a look into these variables individually.

Position 

Position tells us where an object or a quantity is in space. We use the coordinate system to provide an address for a point in space. We generally use the Cartesian coordinate system to describe the position. But we can use other coordinate systems such as the polar coordinate system for two-dimensional space, the cylindrical and spherical polar coordinate system for three-dimensional space. We choose the coordinates according to our convenience.

We know that an object has moved in space when the position coordinate of the object has changed in time. This change in the position will result in the displacement of the body.

This displacement can be described as the vector sum of the change in the position vector.

Displacement does not depend on the path of the motion; it depends only on the final coordinates. But if we travel along a circle, then we reach our initial position in some time. Once we reach that position, the total displacement will be zero, but that does not mean that we have not travelled. In this case, we use the term distance. Distance depends on the path along which the object has moved. Here in the case of the circular motion, we have the displacement to be zero, but the distance travelled will now become 2πr, r being the radius.

Time

Time is the quantity that defines the direction of a sequence of existence and events. Its direction is given by the change in entropy. Time moves forward in the direction in which entropy increases.

The concept of time is used to find the other variables of motion. If we divide the position vector with time, we get velocity, and if we divide again, we get acceleration. According to Newtonian mechanics, time is absolute – it is the same everywhere. 

Note: According to Einstein’s theory of General Relativity, time is not absolute, but rather, time and space are one thing called spacetime. Time can vary according to the frame of reference we choose.

Velocity

Velocity can be defined as the rate of change of a position vector with respect to a frame of reference. It is a vector quantity. For example, if we have a body whose position vector is given by r, we can get the value of velocity by dividing it by time. 

Therefore, v = r/t 

But this formula can be used only when the rate of change of r is constant in time. For the cases where the rate of change of r is not constant in time, we use the instantaneous value of velocity, which is given by,

v = dr/dt

Here, the position vector is differentiated in time. 

Now, let r = xi + yj + zk

Then the velocity if given by, v = (dx/dt)i + (dy/dt)j + (dz/dt)k

Now lets pull out the magnitude of this vector quantity, 

v = ((dx/dt)2 + (dy/dt)2 + (dz/dt)2)1/2 

This quantity is called the speed of the body. It’s a scalar quantity.

Note: If we want to find the value of average velocity, we find the total displacement in a time period and divide that time period.

Now for instances of circular motion, we use another equation for velocity, which is given by,

v = (dr/dt)r + r(dθ/dt)θ 

Here r and θ are the position coordinates in their polar form. 

Now the value of velocity can be used to define another quantity called moment, which we get by multiplying velocity with the mass of the body. We sometimes use momentum as a variable of motion in Hamiltonian and Lagrangian dynamics.

Now, this value of velocity may not always be constant in time; therefore, the rate of change of velocity will give rise to a quantity called acceleration.

Acceleration

Acceleration can be defined as the rate of change of velocity in time. It’s a vector quantity.

Acceleration can be represented in the Cartesian coordinate system as,

a = (d2x/dt2)i + (d2y/dt2)j + (d2z/dt2)k

This will give us the instantaneous value of acceleration. In order to find the value of average acceleration, we find the change in velocity during a time period and divide it with that time period. This is given by, a = (vf – vi)/Δt

For cases involving circular motion, we use the polar coordinates to find the value of acceleration. This is given by,

a = (d2r/dt2 – r(dθ/dt)2)r + (2(dr/dt)(dθ/dt) + r(d2θ/dt2))θ

If we now multiply acceleration with the mass of the body, we get force according to Newton’s second law.

Note: All the variables given here are measured in an inertial frame of reference. If the frame of reference accelerates, then we will have to modify all the above equations to account for the pseudo forces and their dynamics.

Conclusion

The variables of motion are those variables that are used to represent the equations of motion. They are position, time, velocity, and acceleration. Sometimes, we use position and momentum for equations of motion. We use the coordinate system to represent the position. If we find the rate change of the position coordinate, we get velocity, and if we find the rate of change of velocity, then we get the acceleration.