We are aware of the fact that moving object possesses a quality by virtue of which it exerts a force on anything that tries to stop it. The faster the object is travelling, the harder is to stop it. Similarly, if two objects move with the same velocity, then it is more difficult to stop the massive of the two.
This quality of the moving body was called the quantity of motion of the body, by Newton. This term is now called linear momentum of the body and is defined by the relation.
| Linear momentum = p = m v …………. (3.10) |
In this expression v is the velocity of the mass m. linear momentum is, therefore, a vector quantity and has the direction of velocity.
The SI unit of momentum is kilogram metre per second (kg m ). It can also be expressed as Newton second (N s).
Interesting Information
Throwing a package onto shore from a boat that was previously at rest causes the boat to move out-ward from shore (Newton’s third law).
Momentum and Newton’s Second Law of Motion |
Consider a body of mass m moving with an initial velocity vi. Suppose an external force F acts upon it for time t after which velocity becomes vf. The acceleration a produced by this force is given by
By Newton’s second law, the acceleration is given as
Equating the two expressions of acceleration, we have
Or F x t = m vf – m vi ………….. (3.11) |
Where mvi is the initial momentum and mvf is the final momentum of the body
Point to ponder
Which will be more effective in knocking a bear down.
- A rubber bullet or
- A lead bullet of the same momentum
The equation 3.11 shows that change in momentum is equal to the product of force and the time for which force is the form F = ma, because it can easily be extended to account for changes as the body accelerates when its mass also changes. For example, as a rocket accelerates, it loses mass because its fuel is burnt and ejected to provide greater thrust.
Thus, second law of motion can also be stated in terms of momentum as follows
Time rate of change of momentum of a body equals the applied force.
Point to Ponder
Which hurt you in the above situations (a) or (b) and think why?
Impulse |
Sometimes we wish to apply the concept of momentum to cases where the applied force is not constant, it acts for very short time. For example, when a bat hits a cricket ball, the force certainly varies from instant to instant during the collision. In such cases, it is more convenient to deal with the product of force and time (F x t) instead of either quantity alone. The quantity F x t is called the impulse of the force, where F can be regarded as the average force that acts during the time t. from Eq. 3.11
Point to Ponder
Does a moving object have impulse?
Impulse = F x t = m vf – m vi …………… (3.11) |
Example 3.2: A 1500 kg car has its velocity reduced from 
3.0 s. How large was the average retarding force?
Solution: Using the Eq 3.11
The negative sign indicates that the force is retarding one.
Do You Know?
Your hair acts like a crumple zone on your skull. A force of 5N might be enough to fracture your naked skull (cranium), but with a covering of skin and hair, a force of 50 N would be needed.
Law of conservation of momentum: Let us consider an isolated system. It is a system on which no external agency exerts any force. For example, the molecules of a gas enclosed in a glass vessel at constant temperature constitute an isolated system. The molecules can collide with one another because of their random motion but, being enclosed by glass vessel, no external agency can exert a force on them.
Consider an isolated system of two smooth hard interacting balls of masses m1 and m2, moving along the same straight line, in the same direction, with velocities v1 and v2 respectively. Both the balls collide and after collision, ball of mass m1 moves with velocity v’1 and m2 moves with velocity v’2 in the same direction as shown in fig 3.8.
To find the change in momentum of mass m1, using Eq 3.11we have,
Since the action force F is equal and opposite to the reaction force F’, we have F’ = -F, so the left hand side of the equation is zero. Hence,
In other words, change in momentum of 1st ball + change in momentum of the 2nd ball = 0
Which means that total initial momentum of the system before collision is equal to the total final momentum of the system after collision. Consequently, the total change in momentum of the isolated two ball system is zero.
For such a group of objects, if one object within the group experiences a force, there must exist an equal but opposite reaction force on some other object in the same group. As a result, the change in momentum of the group of objects as a whole is always zero. This can be expressed in the form of law of conservation of momentum which states that
The total linear momentum of an isolated system remains constant.
Point to ponder
“What is the effect on the speed of a fighter plane chasing another when it opens fire? What happens to the speed of pursued plane when it returns the fire?”
In applying the conservation law, we must notice that the momentum of a body is a vector quantity.
Example 3.3: two spherical balls of 2.0 kg and 3.0 kg masses are moving towards each other with velocities of 6.0ms-1 respectively. What must be the velocity of the smaller ball after collision, if the velocity of the bigger ball is 3.0 ms-1?
Solution: As both the balls are moving towards one another, so their velocities are of opposite sign. Let us suppose that the direction of motion of 2 kg ball is positive and that of the 3 kg is negative.
Do you wear seat belts?
Do you know?
