Linear Motion, Physics tutorial

Introduction:

Motion may be described as the continuous change of position with time. During motion, different points in the body move along different paths.

Let for simplicity we shall consider motion of a very small body which we shall refer to as a particle. Position of the particle is stated by its projections onto three areas of the Cartesian coordinate system. As particle moves along any path in space, its projections move in straight lines along the three axes. Actual motion can be reconstructed from motions of the three projections.

Motion in a Straight Line and Parameters for describing Motion:

This explains motion in the straight line and parameters for describing motion. These are velocity, displacement and acceleration.

Displacement:

A car which takes 2 hours to travel from Lokoja to Abuja along the winding road, a distance of 200km is said to have the average speed provided by

Speed= distance/time= 200/2hrs= 100kmh-1

An object changes its position at a uniform rate without reference to its direction. In other words, speed is what we call a scalar quantity. Also if a satellite revolves round the earth covering a circular path 60,000km in 24 hours its average speed is:

Average speed= 60,000km/24h= 2500kmh-1

But if satellite moves through equal distance in equivalent times, no matter how small the time intervals the satellite is said to have a constant or uniform speed.

Velocity:

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

Vaverage = v = (x2-x1)/(t2-t1) = Δx/ Δt

Where: x1 is initial distance, x2 final distance.

t1 is initial time, t2 is final time.

The units for velocity can be implied from definition to be meters/second or in general any distance unit over any time unit.

You can approach the expression for instantaneous velocity at any point on the path by taking limit as time interval gets smaller and smaller. Such limiting process is known as a derivative and instantaneous velocity can be stated as

Vinstantaneous = limΔt→0 Δx/Δt= dx/dt

Instantaneous velocity:

The velocity of the particle at someone instant of time, or at some one point of its path, is known as its instantaneous velocity. Average velocity is related with complete displacement and complete time interval. Average velocity could be calculated for very small time intervals- until the limiting time interval is reached. This limiting time interval refers to as the instant of time. Hence instantaneous velocity can be defined as:

V = limΔt→0 Δx/Δt

Instantaneous velocity is also a vector quantity whose direction is limiting direction of displacement vector. By convention, the positive velocity points out that it is towards right along x-axis of coordinate system.

The instantaneous velocity at any point of the coordinate-time graph thus equals slope of tangent to graph at that point.

As the rule of thumb, if tangent slopes upwards to right, its slope is positive, velocity is positive, and motion is to the right. But if tangent slopes downwards to the right, the velocity is negative. At the point where tangent is horizontal, its slope is zero and its velocity is zero. If distance is given in meters and time in seconds, velocity is stated in meters per second (m s-1). Other common units of velocity are:

Feet per second (fts-1) centimeters per second(cms-1) miles per hour (mih-1) and knot (1 knot = 1 nautical mile per hour).

Acceleration: Average and instantaneous acceleration:

When velocity of the object changes it is said to be accelerating. Acceleration is rate of change of velocity with time.

Any change in velocity of the object results in the acceleration: increasing speed also called acceleration, decreasing speed also known as deceleration or retardation, or changing direction. The change in direction of motion results in the acceleration even if moving object neither sped up nor slowed down. That is because acceleration depends on change in velocity and velocity is the vector quantity - one with both magnitude and direction. Therefore, falling apple accelerates, car stopping at the traffic light accelerates, and the orbiting planet accelerates. Acceleration takes place anytime the object's speed increases or decreases, or it changes direction.

There are two types of acceleration: average and instantaneous. Average acceleration is found over a long time interval. Word long in this perspective signifies finite something with the starting and an end. Velocity at starting of the interval is known as initial velocity, symbolized by symbol v0, and velocity at the end is known as final velocity, symbolized by symbol v. Average acceleration is the quantity computed from two velocity measurements.

a¯ = Δv/Δt =  (v - v0) /Δt

In contrast, instantaneous acceleration is estimated over the short time interval. Word short in this context signifies infinitely small or infinitesimal having no period or extent whatever. It is a mathematical best that can only be realized as the limit. Limit of the rate as denominator approaches zero is known as derivative. Instantaneous acceleration is then limit of average acceleration as time interval approaches zero - or alternatively, acceleration is derivative of velocity.

A= limΔt→0 Δv/Δt = dv/dt

Acceleration is a derivative of velocity with time, but velocity is itself a derivative of displacement with time. Derivative is the mathematical operation which can be applied multiple times to a pair of changing quantities. Doing it once provides a first derivative. Doing it twice (the derivative of a derivative) provides second derivative. That makes acceleration first derivative of velocity with time and second derivative of displacement with time.

a = dv/dt = (d/dt) (dx/dt) = d2r/dt2

Rectilinear motion with constant acceleration:

Let us consider body moving in the straight line, if we can observe every particle (which body is composed) is traveling same path, we can describe it as Rectilinear Motion. Or

The body is said to experience rectilinear motion, if two particles in that body travel the equivalent distance along two parallel straight lines. Or

The Motion obtained by the body so that every particle of the body follows the straight line path is expresses as motion that takes place in the straight line.

The parameters involved in the rectilinear motion are displacement, velocity, acceleration and time.

Rectilinear Motion Equations:

Equations for rectilinear motion can be derived from first principle.

Let us suppose that the object begins with the initial velocity 'U' and with the constant acceleration 'a'. The simple calculation states that after time't', increase in velocity during that time t is acceleration × time, i.e. 'at'. As object already had the initial velocity u at the start, the quantity is also added to increase in velocity to find out actual final velocity 'v'.

In other words, first equation for final velocity is stated as,

v = u + at

Now, estimate the final velocity of the object after certain displacement, instead of, after certain time.

Let us suppose that object has traveled the distance of 'S' in a time't'. As we are describing linear motion, distance traveled is provided by,

S = average velocity during time t2.

= 1/2 (u + v) × t................... Eq.1

From Eq.1derived, we can solve for t as,

t = (v-u)/a

Putting this for t in equation,

S = (1/2) (u + v) (v-u)/a, which becomes as,

2aS = (v2 - u2) or,

v2 - u2 = 2aS.

This is another equation for final velocity in terms of initial velocity and distance.

Now, let us deduce the equation for displacement in terms of initial velocity, time and acceleration.

S = Average velocity during time t2

= 1/2 (u + v) × t

Since v = u + at,

S = 1/2 [u +(u + at)] × t

= (1/2) (2u + at)(t).

or,

S = ut + (1/2) at2

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