Understanding Instantaneous Velocity: Calculating Velocity at a Specific Moment in Time using Calculus

instantaneous velocity

Instantaneous velocity refers to the velocity of an object at a specific moment in time

Instantaneous velocity refers to the velocity of an object at a specific moment in time. It is a measure of how fast and in which direction an object is moving at an exact point in time.

To calculate instantaneous velocity, we use the concept of calculus and the derivative. The derivative of displacement with respect to time gives us the instantaneous velocity.

Let’s say we have the equation of motion for an object:
s(t) = 5t^2 + 3t + 2

Here, s(t) represents the displacement of the object at time ‘t’. To find the instantaneous velocity at a specific time, we need to find the derivative of this equation.

To find the derivative, we can use the power rule of differentiation, which states that for any function of the form f(x) = ax^n, the derivative is given by f'(x) = nax^(n-1).

Applying the power rule to our equation, we differentiate each term:

s'(t) = d/dt (5t^2) + d/dt (3t) + d/dt (2)

The derivative of 5t^2 with respect to t is calculated as:
d/dt (5t^2) = 10t

Similarly, the derivative of 3t is:
d/dt (3t) = 3

Since the derivative of a constant is always zero, the derivative of 2 is:
d/dt (2) = 0

Combining the derivatives, we get:

s'(t) = 10t + 3

The resulting equation, s'(t) = 10t + 3, represents the instantaneous velocity of the object at any given time ‘t’ according to the original equation of motion.

To find the instantaneous velocity at a specific time, substitute the value of ‘t’ into the equation. For example, if we want to find the instantaneous velocity at t = 2, we substitute this value into the equation:

s'(2) = 10(2) + 3 = 20 + 3 = 23

Therefore, the instantaneous velocity of the object at time t = 2 is 23 units per time (e.g., meters per second if the units of measurement for displacement were meters and time was seconds).

It’s important to note that instantaneous velocity is a vector quantity, meaning it includes both magnitude (speed) and direction. The result of the calculation above is for the magnitude of velocity. To include direction, we would need additional information, such as the sign convention for velocity or the directional context of the problem.

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