A particle moves along a straight line with equation of motion s=f(t) . where s is measured in m

A particle moves along a straight line with equation of...

Key Concepts

Equation of Motion

The equation of motion describes the position of a particle as a function of time. It encapsulates the particle's path and is fundamental for analyzing motion in kinematics, linking position, displacement, velocity, and acceleration.

Differentiation

Differentiation is the process of finding the derivative of a function. In the context of motion, differentiating the position function with respect to time yields the velocity function, representing the rate at which position changes.

Velocity

Velocity is a vector quantity that represents the rate of change of position with respect to time, including both magnitude and direction. It is calculated by taking the derivative of the position function.

Speed

Speed is the scalar quantity derived from the magnitude of velocity. It measures how fast an object is moving regardless of its direction, and is obtained by taking the absolute value of the velocity.

Transcript

00:01 This problem.

00:01 Number 43 of the steward calculus, eighth edition, section 2.7.

00:05 Particle moves along a straight line with equations of motion, equation of motion as equals f f t, where s is measured in meters and he is is in second find the velocity and speed when t is equal to four.

00:18 So if f represents s and s is measured in meters, it is a position function, and velocity and speed are related to position by a derivative.

00:29 So here we use this definition of the derivative and first attempt to find the derivative of f or the velocity function and then use it to calculate the velocity at t equals to four seconds.

00:43 So f prime of t is, uh, the limited each purchase zero of the function half evaluated at t plus h which will just be this minus the function evaluated at t or just the function itself to minus 18 and then minus negative 60 or plus 60 squared and it's all divided by h.

01:18 Our next step will be to expand the terms in the numerator.

01:27 We have a t t plus 80 h minus t plus h squared that is t squared plus two th plus h squared.

01:37 So this will expand to be negative.

01:39 60 square minus 12 th and a six h squared minus 80 t plus 60 squared.

01:54 Okay, this is all over h now, we, um look for an opportunity to cancel some terms.

02:03 Positivity and negativity.

02:05 T think it is 60 squared and positive.

02:08 60 squared...