Neil AminBrent AndersonPeriod 2AP Physics

Instantaneous Velocity

Definition Velocity, or the motion with a numerical

value and direction, at a specific moment in time.

The difference between velocity and speed is that velocity has a speed with a vector while speed just has speed.

First Example

Problem: A bullet fired in space is traveling in a straight line and its equation of motion is given by S = 4t + 6t2. If it travels for 15 seconds before impact, find the instantaneous velocity at the 10th second.

Solution : We know the equation of motion given by :

S = 4t + 6t 2

The instantaneous velocity is given by : dS / dT (t = 10). i.e. to calculate the instantaneous velocity you must calculate the derivative of displacement with respect to time and substitute t = 10.

dS/dt = d/dt (4t + 6t 2) = 4 + 12t

Therefore,

V Instantaneous at (t =10) = 4 + (12 x 10) = 124 meters/sec.

That bullet is traveling at a phenomenal speed apparently!

How to Find it on a GraphMotion can be found by looking at a graph for position. When the position is going in a positive direction on the graph, there is a positive velocity. When the position is going in a negative direction, the velocity is negative.

Review Table 2.1: Position and time for a runner. t(s) x(m) 1.00 1.00 1.01 1.02 1.10 1.21 1.20 1.44 1.50 2.25 2.00 4.00 3.00 9.00 Find the runner's instantaneous velocity at t = 1.00 s.

Review ContinuedWe can interpret the instantaneous velocity

graphically as follows. Recall that the average velocity is the slope of the line joining P and Q To get the instantaneous velocity we need to take $\Delta$t $\rightarrow$ 0, or P $\rightarrow$ Q. When P $\rightarrow$ Q, the line joining P and Q approaches the tangent to the curve at P (or Q). Thus the slope of the tangent at P is the instantaneous velocity at P. Note that if the trajectory were a straight line, we would get v = $\bar{v}$ , the same for all t .

Questions

How could instantaneous velocity be utilized in a real world application?