Page 103 - Understanding NCERT Science 09

its velocity changes, that is, the object gets an on the time rate at which the momentum is
acceleration. We would now like to study how changed.
the acceleration of an object depends on the The second law of motion states that the
force applied to it and how we measure a force. rate of change of momentum of an object is
Let us recount some observations from our proportional to the applied unbalanced force
everyday life. During the game of table tennis in the direction of force.
if the ball hits a player it does not hurt him.
On the other hand, when a fast moving cricket 8.4.1 MATHEMATICAL FORMULATION OF
ball hits a spectator, it may hurt him. A truck SECOND LAW OF MOTION
at rest does not require any attention when
parked along a roadside. But a moving truck, Suppose an object of mass, m is moving along
–1
even at speeds as low as 5 m s , may kill a a straight line with an initial velocity, u. It is
person standing in its path. A small mass, uniformly accelerated to velocity, v in time, t
such as a bullet may kill a person when fired by the application of a constant force, F
from a gun. These observations suggest that throughout the time, t. The initial and final
the impact produced by the objects depends momentum of the object will be, p = mu and
1
on their mass and velocity. Similarly, if an p = mv respectively.
2
object is to be accelerated, we know that a The change in momentum ∝ p – p
1
2
greater force is required to give a greater ∝ mv – mu
velocity. In other words, there appears to exist ∝ m × (v – u).
some quantity of importance that combines
the object’s mass and its velocity. One such m ( ´ v -u )
The rate of change of momentum ∝
property called momentum was introduced by t
Newton. The momentum, p of an object is
Or, the applied force,
defined as the product of its mass, m and
velocity, v. That is, F ∝ m ( ´ v -u )
p = mv (8.1) t

Momentum has both direction and F = km ( ´ v -u )
magnitude. Its direction is the same as that t (8.2)
of velocity, v. The SI unit of momentum is = k m a (8.3)
kilogram-metre per second (kg m s ). Since
-1
the application of an unbalanced force brings Here a [ = (v – u)/t ] is the acceleration,
a change in the velocity of the object, it is which is the rate of change of velocity. The
therefore clear that a force also produces a quantity, k is a constant of proportionality.
change of momentum. The SI units of mass and acceleration are kg
-2
Let us consider a situation in which a car and m s respectively. The unit of force is so
with a dead battery is to be pushed along a chosen that the value of the constant, k
-1
straight road to give it a speed of 1 m s , which becomes one. For this, one unit of force is
is sufficient to start its engine. If one or two defined as the amount that produces an
persons give a sudden push (unbalanced force) acceleration of 1 m s -2 in an object of 1 kg
to it, it hardly starts. But a continuous push mass. That is,
over some time results in a gradual acceleration 1 unit of force = k × (1 kg) × (1 m s ).
-2
of the car to this speed. It means that the change
of momentum of the car is not only determined Thus, the value of k becomes 1. From Eq. (8.3)
by the magnitude of the force but also by the
F = ma (8.4)
time during which the force is exerted. It may
then also be concluded that the force necessary The unit of force is kg m s -2 or newton,
to change the momentum of an object depends which has the symbol N. The second law of

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