The displacement (in meters) of a particle moving in a...

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

Key Concepts

Evaluating Functions at Specific Points

After obtaining a derivative, evaluating it at particular values of the variable gives the rate of change at those points. In the context of the problem, this means calculating the velocity at predetermined times by substituting those time values into the derivative function.

Power Rule

The power rule is a fundamental rule in differentiation used for functions of the form t^n. It states that the derivative of t^n is n*t^(n-1). This rule simplifies the process of differentiating polynomial expressions, including those that involve negative exponents.

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function changes in response to changes in its input. In physics, it is used to determine the instantaneous rate of change, such as the velocity of an object from its displacement function.

Velocity as the Derivative of Displacement

In kinematics, velocity is defined as the derivative of the displacement with respect to time. By differentiating the displacement function, one obtains the velocity function, which allows for evaluation of the object's speed and direction at any given moment.

Transcript

00:01 This problem number fifteen of this tour calculus eighth edition, section two point seven.

00:06 The displacement and meters of a particle moving in a straight line is given by the equation of motion as equals one over t squared ortiz measured in seconds.

00:15 Find the velocity of the particle times t equals a tickles.

00:18 One tickles too and t equals three.

00:22 So for this for this problem, the position functioned.

00:26 The displacement is given here we know dysfunction.

00:30 The velocity will be the slope of this function of slope function.

00:34 I mean, we'll use definition one where are slow.

00:39 We're gonna call our velocity and its element as t approaches a this is the first part we're going to do thi is equal today parties of progeny out of the function f fifty minus the function evaluated a today divided by the difference of tina our function f ft is given as one over t squared and dysfunction.

01:10 Evaluated a is one of a squared all over eighteen minutes saying we can get rid of the functions in the numerator by multiplying by the that was common denominator, which is? he scored a squared.

01:25 So we're gonna want to apply that to the top of the bottom t squared a squared multiplied by the first term.

01:31 Just he was in a squared.

01:34 He scored a squared multiplied by the second term just leaves a t's great! and in the denominator we'll have this term t minus saying, provided by this t squared a squirter.

01:48 Our next step is too factor thie greater.

01:53 It's a difference of squares, meaning that hey, screw my c squared is the same as a minus.

01:59 Team a plus team on the right, by the quantity t minus a ah t squared a squared.

02:15 Now we can cancel here we can replace a minus t with negative team, anything that's equivalent, and then we can cancel team and with t minus a.

02:29 Those two terms are equivalent as t approaches a and we're left with negative...

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