<< Chapter < Page Chapter >> Page >

v a = Δ s Δ t

On a distance time plot, average speed is equal to the slope of the the straight line, joining the end points of the motion, makes with the time axis. Note that average speed is equal to the slope of the chord (AB) and not that of the tangent to the curve.

Distance time plot

Distance is equal to the tangent of the angle that the chord of motion makes with time axis.

v a = tan θ = Δ s Δ t = BC AC

Average speed

Problem : The object is moving with two different speeds v 1 and v 2 in two equal time intervals. Find the average speed.

Solution : The average speed is given by :

v a = Δ s Δ t

Let the duration of each time interval is t. Now, total distance is :

Δ s = v 1 t + v 2 t

The total time is :

Δ t = t + t = 2t

v a = v 1 t + v 2 t 2t

v a = v 1 + v 2 2

The average speed is equal to the arithmetic mean of two speeds.

Got questions? Get instant answers now!

The object is moving with two different speeds v 1 and v 1 in two equal intervals of distances. Find the average speed.

Solution :

The average speed is given by :

v = Δ s Δ t

Let "s" be distance covered in each time interval t 1 and t 2 . Now,

Δ s = s + s = 2s

Δ t = t 1 + t 2 = s v 1 + s v 2

v a = 2 v 1 v 2 v 1 + v 2

The average speed is equal to the harmonic mean of two velocities.

Got questions? Get instant answers now!

Instantaneous speed (v)

Instantaneous speed is also defined exactly like average speed i.e. it is equal to the ratio of total distance and time interval, but with one qualification that time interval is extremely (infinitesimally) small. This qualification of average speed has important bearing on the value and meaning of speed. The instantaneous speed is the speed at a particular instant of time and may have entirely different value than that of average speed. Mathematically,

v = Δ s 0 Δ s Δ t = d s d t

Where Δ s is the distance traveled in time Δ t .

As Δ t tends to zero, the ratio defining speed becomes finite and equals to the first derivative of the distance. The speed at the moment ‘t’ is called the instantaneous speed at time ‘t’.

On the distance - time plot, the speed is equal to the slope of the tangent to the curve at the time instant ‘t’. Let A and B points on the plot corresponds to the time t and t + Δ t during the motion. As Δ t approaches zero, the chord AB becomes the tangent AC at A. The slope of the tangent equals ds/dt, which is equal to the instantaneous speed at 't'.

Speed time plot

Instantaneous speed is equal to the slope of the tangent at given instant.

v = tan θ = DC AC = ds dt

Speed - time plot

The distance covered in the small time period dt is given by :

ds = v dt

Integrating on both sides between time intervals t 1 and t 2 ,

s = t t 1 t 2 v

The right hand side of the integral represents an area on a plot drawn between two variables, speed (v) and time (t). The area is bounded by (i) v-t curve (ii) two time ordinates t 1 and t 2 and (iii) time (t) axis as shown by the shaded region on the plot.

Area under v-t plot

Area under v-t plot

Area under v-t plot gives the distance covered by the object in a given time interval.

Alternatively, we can consider the integral as the sum of areas of small strip of rectangular regions (vxdt), each of which represents the distance covered (ds) in the small time interval (dt). As such, the area under speed - time plot gives the total distance covered in a given time interval.

Position - time plot

The position – time plot is similar to distance – time plot in one dimensional motion. In this case, we can assess distance and speed from a position - time plot. Particularly, if the motion is unidirectional i.e. without any reversal of direction, then we can substitute distance by position variable. The example here illustrates this aspect of one-dimensional motion.

Problem The position – time plot of a particle’s motion is shown below.

Position - time plot

Determine :

  • average speed in the first 10 seconds
  • Instantaneous speed at 5 seconds
  • Maximum speed speed
  • The time instant(s), when average speed measured from the beginning of motion equals instantaneous speed

Solution

1. Average speed in the first 10 seconds

The particle covers a distance of 110 m. Thus, average speed in the first 10 seconds is :

v a = 110 10 = 11 m/s

2. Instantaneous speed at 5 second

We draw the tangent at t = 5 seconds as shown in the figure. The tangent coincides the curve between t = 5 s to 7 s. Now,

Position - time plot

v = slope of the tangent = 30 2 = 15 m/s

3. Maximum speed

We observe that the slope of the tangent to the curve first increase to become constant between A and B. The slope of the tangent after point B decreases to become almost flat at the end of the motion. It means the maximum speed corresponds to the constant slope between A and B, which is is 15 m/s

4. Time instant(s) when average speed equals instantaneous speed

Average speed is equal to slope of chord between one point to another. On the other hand, speed is equal to slope of tangent at point on the plot. This means that slope of chord between two times is equal to tangent at the end point of motion. Now, as time is measured from the start of motion i.e. from origin, we draw a straight line from the origin, which is tangent to the curve. The only such tangent is shown in the figure below. Now, this straight line is the chord between origin and a point. This line is also tangent to the curve at that point. Thus, average speed equals instantaneous speeds at t = 9 s.

Position - time plot

Got questions? Get instant answers now!

Problem : Two particles are moving with the same constant speed, but in opposite direction. Under what circumstance will the separation between two remains constant?

Solution : The condition of motion as stated in the question is possible, if particles are at diametrically opposite positions on a circular path. Two particles are always separated by the diameter of the circular path. See the figure below to evaluate the motion and separation between the particles.

Motion along a circular path

Two particles are always separated by the diameter of the circle transversed by the particles.

Got questions? Get instant answers now!

Questions & Answers

What are the system of units
Jonah Reply
A stone propelled from a catapult with a speed of 50ms-1 attains a height of 100m. Calculate the time of flight, calculate the angle of projection, calculate the range attained
Samson Reply
58asagravitasnal firce
Amar
water boil at 100 and why
isaac Reply
what is upper limit of speed
Riya Reply
what temperature is 0 k
Riya
0k is the lower limit of the themordynamic scale which is equalt to -273 In celcius scale
Mustapha
How MKS system is the subset of SI system?
Clash Reply
which colour has the shortest wavelength in the white light spectrum
Mustapha Reply
how do we add
Jennifer Reply
if x=a-b, a=5.8cm b=3.22 cm find percentage error in x
Abhyanshu Reply
x=5.8-3.22 x=2.58
sajjad
what is the definition of resolution of forces
Atinuke Reply
what is energy?
James Reply
Ability of doing work is called energy energy neither be create nor destryoed but change in one form to an other form
Abdul
motion
Mustapha
highlights of atomic physics
Benjamin
can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
أوك
عباس
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply

Get the best Physics for k-12 course in your pocket!





Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics for k-12' conversation and receive update notifications?

Ask