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Many modern technologies, such as computers and smartphones, are built on the principles of Physics Topics such as quantum mechanics and information theory.
What are 5 Differences Between Speed and Velocity?
Definition: The rate of displacement of a body with time is called its velocity.
In other words, the rate of change of position of any object with respect to time is its velocity.
The change of position, i.e., the displacement is a vector quantity. If s is the displacement of an object in time t, then,
velocity (v) \(=\frac{\text { displacement }(s)}{\text { time }(t)}\)
Velocity, like displacement, is also a vector quantity.
Units and dimension of velocity: Since the units of distance covered and that of displacement are the same, the units of speed and velocity are also the same,
Dimension of velocity \(=\frac{\text { dimension of displacement }}{\text { dimension of time }}\)
= \(\frac{\mathrm{L}}{\mathrm{T}}\) = LT-1
Therefore, the dimension of velocity is also identical to that of speed.
Uniform and non-uniform velocity: If the velocity of a particle has a constant magnitude and a constant direction it is called uniform velocity. On the other hand, if the velocity of a particle changes with time, either in magnitude or in direction or in both, it is termed as a non uniform velocity.
Due to gravity, the velocity of a falling body increases in magnitude keeping its direction unchanged. Therefore, the velocity of the body is non-uniform. Again, a car moving with a constant speed along a curved path has a non-uniform velocity due to its continuous change in direction.
A uniform circular motion is an example of a motion with uniform speed but non-uniform velocity. [see details in the chapter Circular Motion]
Average velocity:
Average velocity, (v) \(=\frac{\text { total displacement }(s)}{\text { total time }(t)}\)
i.e., dividing the total displacement of a particle in a certain interval of time by the time interval, its average velocity is obtained.
‘A stone takes 4 s to reach the ground when dropped from a height of 80 m ‘—this statement provides no information about the change in velocity of the stone along the path. But it can be said that the average downward displacement of the stone per second is \(\frac{80}{4}\) or 20 m. So the average velocity of the stone is 20 m ᐧ s-1.
Instantaneous velocity: Velocity of a particle at any moment is called its instantaneous velocity. The instantaneous velocity can be defined similarly as the instantaneous speed.
Definition: The instantaneous velocity of a particle at a given point is the limiting value of the rate of displacement from that point with respect to time when the time interval tends to zero.
Following the rule of differential calculus, if Δt is the time in which the displacement of any particle is Δs, then instantaneous velocity is
vi = \(\lim _{\Delta t \rightarrow 0} \frac{\Delta s}{\Delta t}\) = \(\frac{d s}{d t}\)
where, s is the displacement of the particle from the given point.
Comparison between average velocity and instantaneous velocity:
i) Average velocity \(=\frac{\text { total displacement }}{\text { total time }}\)
But instantaneous velocity = \(\frac{d s}{d t}\)
Indeed, the average velocity in an infinitesimally small interval of time is called the instantaneous velocity.
ii) The instantaneous velocity becomes equal to the average velocity of a particle only if it moves with a uniform velocity. Otherwise we cannot get any idea about the instantaneous velocities at different points from the average velocity of a particle.
iii) In kinematics, ideas about the equality of velocities, the change in velocity, etc. are very important. The knowledge of the average velocity alone does not give any idea about them. Thus, the concept of instantaneous velocity is more important.
Comparison between speed and velocity
i) Rate of distance travelled with time is speed whereas rate of displacement with time is velocity.
ii) The units and dimension of distance travelled and those of displacement are the same. So, the units and dimension of speed and those of velocity are the same.
iii) Speed is a scalar quantity, but velocity is a vector quantity.
iv) An object moving along a straight line with uniform speed has a uniform velocity as well, i.e., uniform velocity means a uniform speed in a fixed direction.
v) Uniform velocity always indicates uniform speed, but the converse is not true. A body moving with uniform speed in a curved path has a non-uniform velocity due to change in its direction.
vi) Speed is always positive or zero, but velocity may also be negative depending on the direction of motion.
vii) Average speed of an object is zero means that average velocity is zero too but the converse may not be true always.
viii) Instantaneous speed and instantaneous velocity at any point of motion are the same in magnitude and independent of the shape of the path. But if an object follows a curved path, its average speed and average velocity at any interval of time are different in magnitude.
Numerical Examples
Example 1.
A particle moves in a circular path of radius 7 cm. It covers
(i) half of the circle in 4 s and
(ii) one complete round in 10 s. In each case find the average speed and average velocity.
Solution:
The circumfèrence of the circle
= 2πr = 2 × \(\frac{22}{7}\) × 7 = 44 cm
i) The distance travelled by the particle = half of the circumference = \(\frac{22}{4}\) = 22 cm and time taken = 4 s.
∴ Average speed, v = \(\frac{22}{4}\) = 5.5 cm s
Displacement diameter of the circle = 2 × 7 = 14 cm and time taken = 4 s.
Hence average velocity, \(\vec{v}\) = \(\frac{14}{4}\) = 3.5 cm ᐧ s-1 along AB.
ii) The distance travelled by the particle = circumference of the circle = 44 cm and time taken = 10 s.
∴ Average speed = \(\frac{44}{10}\) = 4.4 cm ᐧ s-1.
Displacement in this case is 0 (as initial and final positions are the same.)
Hence, average velocity displacement \(=\frac{\text { displacement }}{\text { time }}\) = 0
Example 2.
An aeroplane travels 2000 km to the west. It then turns north and moves 2000 km more. Finally it follows the shortest path to return to its starting point. If the speed of the plane is 200 km ᐧ h-1, find its average velocity for the total journey.
Solution:
Since initial and final positions are the same, displacement is zero.
∴ Average velocity \(=\frac{\text { displacement }}{\text { total time }}\) = \(\frac{0}{\text { total time }}\) = 0
Example 3.
Find the speed of the tip of a 3 cm long second’s hand in a clock.
Solution:
The tip of the second’s hand describes an angle of 360° in 60 seconds when it completes the total circular path once.
Hence, distance travelled = circumference of the circle = 2π × 3 = 6π cm, time = 60 s.
∴ Speed = \(\frac{6 \pi}{60}\) = \(\frac{\pi}{10}\) = 0.314 cm ᐧ s-1
Example 4.
A train travels from station A to station B at a constant speed of 40 km ᐧ h-1 and returns from B to A at 60 km ᐧ h-1. Find the average speed and average velocity of the train.
Solution:
Let the distance between the stations A and B be x km.
Time taken by the train to move from A to B = \(\frac{x}{40}\)h and time taken to move from B to A = \(\frac{x}{60}\)h.
Total distance travelled = 2x km.
∴ Average speed = \(\frac{2 x}{\left(\frac{x}{40}+\frac{x}{60}\right)}\) = \(\frac{2 \times 40 \times 60}{100}\) = 48 km ᐧ h-1.
Initial and final positions are the same and so the total displacement becomes zero and hence average velocity is zero.
Example 5.
The motion of a particle, along x – axis, follows the relation x = 8t – 3t2. Here x and t are expressed in metre and second respectively. Find
(i) the average velocity of the particle in time interval 0 to 1 s and
(ii) its instantaneous velocity at t = 1 s.
Solution:
i) Let at t = 0, x = x1 and at t = 1, x = x2.
∴ x1 = 8 × 0 – 3 × 02 = 0
and x2 = 8 × 1 – 3 × 12 = 5 m
∴ Displacement x2 – x1 = -5 m and time taken is 1s.
∴ Average velocity = \(\frac{5 \mathrm{~m}}{1 \mathrm{~s}}\) = 5 m ᐧ s-1.
ii) Here instantaneous velocity
v = \(\frac{d x}{d t}\) = \(\frac{d}{d t}\)(8t – 3t2) = 8 – 6 × 1 = 2 m ᐧ s-1.
At t = 1 s, \(\frac{d x}{d t}\) = 8 – 6 × 1 = 2m ᐧ s-1.