Dynamics: Kinematics of a Particle

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Kinematics of a Particle

Length: 15 Minutes 46 Seconds

## Kinematics

By studying kinematics of a particle you will be able to determine the position, velocity, and acceleration of a particle at any given time. Since everything is going to be treated as a particle, rotation about a fixed axis will not be included until the rigid body sections.

## Displacement

Displacement is the particles position at any given time. If the particle is going in a straight line the change in displacement can be expressed by equation 1 and the figure below.

(1)

## Velocity

Velocity is the change in displacement over time. Velocity can be expressed by equations 2.

(2)

## Acceleration

Acceleration is the change in velocity over time. Refer to equation 3.

(3)

If there is a constant acceleration that is being applied, equations 4 and 5 could be used to determine the velocity or displacement due to the constant acceleration. Equation 4 takes in consideration of an initial velocity, and equation 5 takes in consideration of an initial velocity and displacement.

(4)

(5)

## Multi Directional Situations

We live in a 3 dimensional world so in most cases a particle could be traveling in the x, y, and z direction at the same time. The best way to deal with a problem like this is to treat it as three one dimensional problems. Once each one has been solved, the magnitude for displacement, velocity, and acceleration can be found by using equations 6 thru 8.

(6)

(7)

(8)

## Projectile Motion

Projectile motion is the study of how an object moves through the air. A projectile motion problem can be thought of as a 2 dimensional problem with motion in the horizontal direction (x direction) and motion in the vertical direction (y direction). For the horizontal direction the velocity would be constant and that would be used to calculate the position in that direction However, for the vertical direction there is a constant acceleration due to gravity. Which means the gravitational acceleration also has to be considered in the equations to calculate position and velocity. The gravitational constant in SI units is 9.81 m/s^2, while for British-American units it would be 32.2 ft/s^2. To view a general example of projectile motion refer to the example below.

## Normal and Tangential Components

When a particle goes around a curve it develops a tangential component and a normal component. To view the normal and tangential components on a curve refer to the image below.

To calculate the normal and tangential components the following equations would be used.

(9)

(10)

(11)

## Pulleys

Pulleys can be used to provide a mechanical advantage when lifting heavy objects. However, to gain a mechanical advantage a sacrifice has to be made. This sacrifice is that one end of the pulley system will have to move a greater distance of travel then the other end. An example of how to calculate this is shown below.

## Two Particles in Relationship to Each Other

Two particles can be related to each other with a relative position, velocity, or acceleration at an instant in time. To calculate the relative position, velocity, or acceleration the following equations would be used.

(12)

(13)

(14)

In many cases these types of problems will have motion in multiple directions. To solve these types of problems refer to the example below.