Particle

1. Particle A particle is an object having a non zero mass and the shape of a point (zero size and no internal structure). In various situations we approximate a real object by a particle. For the translational motion of an object we can assume that the object is a particle having the mass of the object and placed at the center of mass of the object.

2. Position - a vector quantity associated with a configuration of the universe. z An oriented segment from the reference point to the particle and assigned triad of numbers represent the position of the particle. z r r y O y x x r r r = [x,y,z]

3. displacement z In general, the position of a particle depends on time. r The difference in position (vector) of a particle at two different instances t1 and t2 is called the displacement (vector) of the particle, in the time interval (t1,t2). r(t2) r(t1) r(t) y r = r(t2) – r(t1) x Note that r = r(t2) – r(t1) r = r(t2) – r(t1) and

4. v r(t) dr r(t+dt) Definition of velocity z The rate, at which a particle is changing its position (vector), is called the velocity (vector) of the particle. y x Note that and

5. v(t) v(t+dt) a(t) -v(t) dv v(t+dt) Definition of acceleration The rate, at which a particle is changing its velocity (vector), is called the acceleration (vector) of the particle. z y Note that x and

6. f f f () f (+) derivatives of vectors The derivative of a function f()of one variable , is a function f ’() defined by the following equation 

7. differential of a function The infinitesimal change df in the value of the function f () due to the infinitesimal change d of the argument is called the differential of the function. f() df f  d note that

8. scalar components of a vector derivative Each component of a vector is differentiated separately.

9. v v v v vx vx vx vx a a a a vz vz vz example: projectile motion The general function representing the position of a projectile at instant t is a quadratic function of time: At instant t the value of the velocity z and the acceleration is warning! x Valid only for motion with constant acceleration.

10. Inverse relations The first fundamental theorem of calculus: Let f (t) be a continuous function and F’(t) = f (t), then Hence: If the velocity of the particle is known at instant t1 and the acceleration of the particle is known at all instances t' between t1 and t, at instant t the particle has velocity

11. Inverse relations … and If the position of the particle is known at instant t1 and the velocity of the particle is known at all instances t' between t1 and t, at instant t the particle has the position

12. i f (i) integral of a vector function The definite integral of vector function f() over an [a,b] interval is defined as The definite integral of function f() over an [a,b] interval is defined as  a b i geometric interpretation: the area bound by the plot of the function general (my personal) interpretation: sum of all value of the function in the [a,b] interval

13. Scalar components of a vector integral Each component of a vector is integrated separately.

14. example: motion with constant acceleration - the acceleration of a particle does not depend on time. The velocity of the particle is a linear function of time. Note: (initial velocity) The position of the particle is a quadratic function of time Note: (initial position)

15. dr speed The magnitude of velocity is called speed theorem Speed is equal to the rate at which the particle moves along the path. conclusion The length of the particle's path is equal to the integral of its speed over the time.

16. average values The average value of a function f () over an interval a,b is a number assigned as follows: fav  a b comment

17. r average velocity t1 t2 Ratio of “the displacement over time” results in the average velocity.

18. v average acceleration t1 t2 Ratio of “the change in velocity over time” results in the average acceleration.

19. vector product C The vector product of two vectors A and B is a vectorC, the magnitude of which is C = ABsin (where is the angle between the multiplied vectors), and the direction of which is perpendicular to the plane formed by the multiplied vectors, following the right-hand rule.  A B

20. important theorems the components anticommutitive distributive over addition of vectors differentiation follows the product rule a useful identity

21.  a r  v uniform circular motion A motion with a constant speed along a circular path is called a uniform circular motion. z v   r y x

22. z’ z , , and t = t’ , then if r’ P R O’ y’ r y O x x’ Note that in general Galilean transformation Motion is always relative. The relationship between description of a physical phenomenon in one reference frame with the description in another is called a transformation.