Find velocity, Calculus and rates of change

[physics604]

1. The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s=$\frac{1}{t^2}$, where t is measured in seconds. Find the velocity of the particle at times t=a, t=1, t=2, and t=3.

Homework Equations

v=$\frac{d}{t}$

The Attempt at a Solution

The question says the d=$\frac{1}{t^2}$

I plug that into V=$\frac{d}{t}$, getting s=$\frac{1}{t^3}$.

When I input t=a, 1, 2, and 3, I get $\frac{1}{a^3}$, 1, $\frac{1}{8}$, and $\frac{1}{27}$.

However, the textbook says that the answers are -$\frac{2}{a^3}$, -2, -$\frac{1}{4}$, and -$\frac{2}{27}$.

Where did it get the -2 from? What am I doing wrong?

Thanks in advance.

 

[Mark44]

1. The displacement (in meters) of a particle moving in a straight line is given by the equation of motion s=$\frac{1}{t^2}$, where t is measured in seconds. Find the velocity of the particle at times t=a, t=1, t=2, and t=3.

Homework Equations

v=$\frac{d}{t}$

The Attempt at a Solution

The question says the d=$\frac{1}{t^2}$

I plug that into V=$\frac{d}{t}$, getting s=$\frac{1}{t^3}$.

When I input t=a, 1, 2, and 3, I get $\frac{1}{a^3}$, 1, $\frac{1}{8}$, and $\frac{1}{27}$.

However, the textbook says that the answers are -$\frac{2}{a^3}$, -2, -$\frac{1}{4}$, and -$\frac{2}{27}$.

Where did it get the -2 from? What am I doing wrong?

The formula you are using is the average velocity. In the problem, you're supposed to use the instantaneous velocity, $\frac{ds}{dt}$. Your book should have examples of how to find the derivative of the displacement, s.

 

[physics604]

Thanks! According to my textbook, the equation for

instantaneous rates of change = lim x₂→x₁ $\frac{f(x2)-f(x1)}{x2-x1}$.

In my case would that mean the equation would be

$\frac{Δd}{Δt}$ = lim t2→t1 $\frac{f(d2)-f(d1)}{t2-t1}$ ?

How does that work into the question?

 

[Mark44]

$$v = ds/dt = \lim_{h \to 0}\frac{s(t + h) - s(t)}{h}$$

This formula is equivalent to the one in your book, with x₂ = t + h and x₁ = t.

Your function is s = s(t) = 1/t².

So s(t + h) = ?
And s(t) = ?