Velocity is defined as the rate of change of displacement with respect to time. It is a vector quantity, meaning it has both magnitude and direction. Speed, on the other hand, is the magnitude of velocity and does not take into account the direction of motion.
To understand why velocity is the derivative of speed, let's consider a simple scenario of an object moving along a straight line. Suppose the position of the object at time t is given by x(t). The displacement of the object at any given time interval can be calculated as Δx = x(t2) - x(t1), where t2 and t1 are two different points in time.
The average velocity of the object during this time interval is given by: v_avg = Δx / Δt
where Δt = t2 - t1 is the time interval. If we consider smaller and smaller time intervals, we can approach the instantaneous velocity of the object at a specific point in time.
Now, let's define speed as the magnitude of the velocity: s = |v|
The derivative of a function represents the instantaneous rate of change of that function. In this case, we want to find the derivative of speed (s) with respect to time (t). Mathematically, it is expressed as:
ds/dt = d/dt |v|
To find this derivative, we need to consider the absolute value function |v|. The absolute value of a vector is a scalar quantity, so we can rewrite |v| as √(v^2) using the properties of vectors.
ds/dt = d/dt √(v^2)
Now, applying the chain rule of differentiation, we can simplify the equation further. Let's assume v is a function of time t, so v = v(t).
ds/dt = (1/2)(v^2)^(-1/2) * d/dt (v^2)
ds/dt = (1/2)(v^2)^(-1/2) * 2v * dv/dt
Simplifying the expression:
ds/dt = v * (v^2)^(-1/2) * dv/dt
Since v is the magnitude of velocity (speed), v = s. Therefore, the equation becomes:
ds/dt = s * (s^2)^(-1/2) * dv/dt
Simplifying further:
ds/dt = s * s^(-1) * dv/dt
ds/dt = dv/dt
We end up with ds/dt = dv/dt, which means the derivative of speed with respect to time is equal to the derivative of velocity with respect to time. Therefore, velocity is the derivative of speed.