2.4 Instantaneous velocity and speed

Instantaneous speed

The speed of a particle at any instant of time is known as its instantaneous speed.
\(
\text { Instantaneous speed }=\lim _{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}=\frac{d s}{d t}
\)
where, \(s\) represents distance.

Example 1: The distance travelled by a particle in time \(t\) is given by \(s=\left(2.5 \mathrm{~m} / \mathrm{s}^2\right) t^2\). Find the instantaneous speed at \(t=5 \cdot 0 \mathrm{~s}\).

Solution: 
\(
s=\left(2.5 \mathrm{~m} / \mathrm{s}^2\right) t^2
\)
\(
\frac{d s}{d t}=\left(2.5 \mathrm{~m} / \mathrm{s}^2\right)(2 t)=\left(5.0 \mathrm{~m} / \mathrm{s}^2\right) t
\)
At \(t=5.0 \mathrm{~s}\) the speed is
\(
v=\frac{d \mathrm{~s}}{d t}=\left(5 \cdot 0 \mathrm{~m} / \mathrm{s}^2\right)(5 \cdot 0 \mathrm{~s})=25 \mathrm{~m} / \mathrm{s}
\)

Instantaneous velocity

The velocity of a particle at any instant of time is known as its instantaneous velocity.
\(
v=\lim _{\Delta t \rightarrow 0} \frac{\Delta x}{\Delta t}=\frac{d x}{d t}
\)

Example 2:A ball is thrown upward, and its position is described by the equation \(x(t)=4 t^2+10 t+6\), where \(x\) is in meters and \(t\) is in seconds. What is the instantaneous velocity of the ball at \(t=5\) seconds?

Solution: Find the velocity function. To find the velocity function, take the derivative of the position function with respect to time.
The derivative of \(x(t)=4 t^2+10 t+6\) is \(v(t)=8 t+10\).
Calculate the instantaneous velocity. Substitute the given time into the velocity function.
At \(t=5\) seconds, the instantaneous velocity is \(v(5)=8(5)+10\).
\(v(5)=40+10=50 \mathrm{~m} / \mathrm{s}\).