Constant Velocity Definition
We will try to define the constant velocity from Newtons law of motion. We will repeat the first law which states that when an object is set into motion, it is supposed to continue to be in motion with a constant velocity, unless some external force acts on that. Therefore, it means a constant velocity is possible if that external force is 0. Now as per Newtons second law a force is defined as the product of the mass of the object and the acceleration created. Therefore, if an external force has to be 0, the acceleration must be 0, as the mass can never be zero. Thus, we can define that the velocity of an object can be constant if there is no acceleration affected. In other words, constant velocity is, a motion with zero acceleration.
Constant Velocity Equation
The equation for a constant velocity or a constant velocity formula is very simple. If v is the velocity of an object and if it is constant, the constant velocity formula is, v = k, where k is any constant. We can obviously derive that, if a is the acceleration, a = dvdt = 0. On the other hand the definition of instant velocity is, v = dsdt, where s is the displacement and t is the time. In case of constant velocity, dsdt = k Now integrating both sides with respect to t, s = kt + c, where c is another constant. We note that this is the slope intercept form of a linear equation. Therefore, an object moving with a constant velocity covers equal displacement in equal time intervals. This is another definition of constant velocity of an object in motion.
Constant Velocity Graph
Since the velocity function in case of constant velocity is defined as V = k, where k is any constant, the velocity graph of an object with constant velocity with respect to time will be a horizontal line, as the slope is 0. As the displacement function under constant velocity motion is a linear function, the displacement graph will be a straight line having a slope equal to the velocity of the object and yintercept representing the initial displacement from the reference point. Obviously, because of zero acceleration, x-axis will be the acceleration graph of the motion with constant velocity.
The graph of an example case is shown below :
Constant Acceleration Motion
Constant acceleration motion can be characterized by formuli and by motion graphs.
Calculus Application for Constant Acceleration
The motion equations for the case of constant acceleration can be developed by integration of the acceleration. The process can be reversed by taking successive derivatives.
Calculus Application for Constant Acceleration
The motion equations for the case of constant acceleration can be developed by integration of the acceleration. The process can be reversed by taking successive derivatives. On the left hand side above, the constant acceleration is integrated to obtain the velocity. For this indefinite integral, there is a constant of integration. But in this physical case, the constant of integration has a very definite meaning and can be determined as an intial condition on the movement. Note that if you set t=0, then v = v0, the initial value of the velocity. Likewise the further integration of the velocity to get an expression for the position gives a constant of integration. Checking the case where t=0 shows us that the constant of integration is the initial position x0. It is true as a general property that when you integrate a second derivative of a quantity to get an expression for the quantity, you will have to provide the values of two constants of integration. In this case their specific meanings are the initial conditions on the distance and velocity.
Time Dependent Acceleration
If the acceleration of an object is time dependent, then calculus methods are required for motion analysis. The relationships between position, velocity and acceleration can be expressed in terms of derivatives or integrals.
