1. ## Instantaneous Velocity

A flare is shot up from the deck of a ship, the initial upward velocity of the flare is 30 m/s. The height of the flare is given by $\displaystyle h=-5t^{2}+30t+10$

The Question is:
Determine the instantaneous velocity of the flare at t=3 seconds by using the slopes of the secants.

In other examples I found the instantaneous velocity by creating a table and using numbers that get closer and closer to 3 seconds until I can reach a conclusion.....but how do I use the slopes of the secants to find it?

Thank you

2. Originally Posted by bearhug
A flare is shot up from the deck of a ship, the initial upward velocity of the flare is 30 m/s. The height of the flare is given by $\displaystyle h=-5t^{2}+30t+10$

The Question is:
Determine the instantaneous velocity of the flare at t=3 seconds by using the slopes of the secants.

In other examples I found the instantaneous velocity by creating a table and using numbers that get closer and closer to 3 seconds until I can reach a conclusion.....but how do I use the slopes of the secants to find it?

Thank you
Ideally one would differentiate h and find the instantaneous velocity. Apparently "slope of secants" is the approximate idea of differentiation. So my guess is, "I found the instantaneous velocity by creating a table and using numbers that get closer and closer to 3 seconds until I can reach a conclusion" is called "slope of secants" method.

Lets call the height h(t), since its a function of time, then instantaneous velocity at t=3 is $\displaystyle \frac{h(3 + x) - h(3)}{x}$, where x can be any small number(like 0.01,0.001 etc)

3. Notice that $\displaystyle \frac{h(3+x)- h(3)}{x}$ is the slope of the secant line: the line between (3, h(3)) and (3+x, h(3+x)). Do that calculation and see what happens for very small values of x.