mechanics motion

MechanicsBBT MIC 211 Physics for Biologists /

Mechanics / Motion

BBT MIC 211 Physics for Biologists / Mechanics / Motion


We describe motion in terms of space and time while ignoring the agents that caused that motion.

Motion represents a continuous change in the position of an object.

In physics we are concerned with three types of motion: translational, rotational, and vibrational.

Translational motion:

Example: A car moving down a highway

Rotational motion:

Vibrational motion:

Example: Earth’s spin on its axis

Example: Back-and-forth movement of a pendulum

Motion


Description of Motion in One Dimension

Motion is described in terms of displacement (x), time (t), velocity (v), and acceleration

Position Specifying the position of an object is essential in describing motion. In one dimension some typical ways are

The vector change in position associated with a motion is called the displacement.

A position vector r may be expressed in terms of the unit vectors

In three dimensions, cartesian or spherical polar coordinates are used, as well as other coordinate systems for specific geometries.

Displacement

.

The displacement of an object is defined as the vector distance from some initial point to a final point. It is therefore distinctly different from the distance traveled except in the case of straight line motion. The distance traveled divided by the time is called the speed, while the displacement divided by the time defines the average velocity.

If the positions of the initial and final points are known, then the distance relationship can be used to find the displacement.


Velocity

The average speed of an object is defined as the distance traveled divided by the time elapsed. Velocity is a vector quantity, and average velocity can be defined as the displacement divided by the time. For the special case of straight line motion in the x direction, the average velocity takes the form:

the instantaneous velocity can be defined as

If the beginning and ending velocities for this motion are known, and the acceleration is constant, the average velocity can also be expressed as

For general cases involving non-constant acceleration, this definition must be applied directly because the straight line average velocity expressions do not work.


Acceleration

Acceleration is defined as the rate of change of velocity. Acceleration is inherently a vector quantity, and an object will have non-zero acceleration if its speed and/or direction is changing. The average acceleration is given by

The units for acceleration can be written as m/s2.

The instantaneous acceleration at any time may be obtained by taking the limit of the average acceleration as the time interval approaches zero. This is the derivative of the velocity with respect to time:


Equations of motion

Motion Graphs Motion Graphs Explained

Equations of motion

Constant Acceleration Motion

The motion equations for the case of constant acceleration can be developed by integration of the acceleration. The process can be reversed by taking successive derivatives.


Equations of motion

Time Dependent Acceleration

If the acceleration of an object is time dependent, then calculus methods are required for motion analysis. The relationships between position, velocity and acceleration can be expressed in terms of derivatives or integrals.


kinematics: A description of motion position: your coordinates displacement: x = change of position velocity: rate of change of position

average : x/tinstantaneous: slope of x vs. t

acceleration: rate of change of velocityaverage: v/tinstantaneous: slope of v vs. t

Summary of Concepts

Solution:

Problem # 1

The vertical motion of an object is graphed in Figure. Describe its motion qualitatively, and find its instantaneous velocity at points A, B, and C.

Solution:

Problem # 2

Ans: (a) 40.0m/s, (b) 20.0m/s, (3) 100m

Problem # 3

Solution:

Ans: (a) 31.3 m/s, (b) 3.19s

Problem # 4

Solution:

Ans: (a) -4.0m/s2, (b) 2.0m


Motion in Two Dimensions

Displacement vector is defined as the difference between its final position vector and its initial position vector.

Average velocity of a particle during the time interval t as the displacement of the particle divided by that time interval

A particle moving in the xy plane is located with the position vector r drawn from the origin to the particle.


Two dimensional equation of motion with constant acceleration


Projectile motion is one type of two-dimensional motion under constant acceleration, where ax=0 and ay=-g.

Projectile Motion

The motion should be analyzed in terms of separate horizontal and vertical components.

It is useful to think of projectile motion as the superposition of two motions.

1. Constant-velocity motion in the x direction

2. Free fall motion in the vertical direction subject to a constant downward acceleration of magnitude g=9.8m/s2


A Projectile fired from the origin at ti=0 with an initial velocity vi. The maximum height of the projectile is h, and the horizontal range is R. At point A, the peak of the trajectory, the particle has coordinates (R/2, h) andAt point B the particle has coordinates (R,0)

Equation of Projectile Motion

We have to determine R and h in term of i vi and g.


Problem # 1

Solution:

Ans: (a) 4.04s, (b) 121m, (3) 50 m/s


Problem # 2

Solution:

Ans: 884 m

A baseball is thrown with an initial velocity of 100 m/s at an angle of 30 degree above the horizontal. How far from the throwing point will the baseball attain its original level?


Problem # 3

Solution:

Ans: (a) x=vi2sin2/g , (b) 310

(a) Find the range x of a gun which fires a shell with muzzle velocity v at an angle of elevation .

(b) Find the angle of elevation of a gun which fires a shell with a muzzle velocity of 120 m/s and hits a target on the same level but 1300m distant.


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