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) = ?