## Key Concepts

- Definition of momentum
- Momentum vs. velocity
- Momentum vs. mass
- Expression and units of momentum
- Numerical problems on momentum

**Introduction:**

In our daily life, we make many observations, such as a fast bowler taking a run-up before bowling, a tennis player moving her racket backward before hitting the tennis ball and a batsman moving his bat backward before hitting the cricket ball. All these activities are performed to make the ball move with great speed when hit/thrown. The balls would rather move slowly if these activities are not done. In all these cases, a force is applied to a ball, and that makes it move faster. Such activities are said to provide **momentum** to the ball. The **greater the force** imparted, the **greater the momentum **imparted, the **faster** the object moves.

**Explanation:**

### Momentum

It is defined as the amount of motion contained in an object. It shows the fastness of motion of an object or the strength of the impact that the moving object would cause. It is denoted by the letter **p**.

### Momentum vs. velocity:

Suppose a ball is thrown by a child and his father, The ball thrown by the father would move to a greater distance because he is stronger than his child. This difference in the impact (the distance moved by the ball) in each case is different because of the difference in **velocity** acquired by the ball. When thrown by the child, the ball moves with a much lesser speed than the speed it acquires when thrown by his father. Thus, the ball thrown by the father has **greater momentum** than the same ball thrown by the child. Therefore, it can be concluded that the momentum of an object increases with increasing speed, i.e., **momentum varies directly as the speed**.

Mathematically, **p α v**

It means that the momentum of an object increases when its speed increases.

Suppose there is a door made up of toughened glass and one needs to break it. What would one choose to break it, a larger stone or a smaller marble? Figure 7.1 is given below for reference.

When the stone is thrown on the door, the toughened glass door will break. On the other hand, if the small marble is thrown instead, the toughened glass door might not break. This is because the momentum of the stone is greater than that of the marble, which is of a greater mass as compared to the stone. The only difference between the stone and the marble here is their masses, making them have a different impact on the glass door.

Thus, momentum increases with an increase in mass of the moving body, i.e., the **momentum of a body varies directly as its mass**.

Mathematically, ** p α m**

It means that the momentum of an object increases when its speed increases.

Further, it is possible to use the smaller marble and get the same impact, which can be caused by the larger stone breaking the toughened glass door. The marble has a **small mass**, unlike the stone. However, it can also break the toughened glass if its momentum is somehow increased. One way to increase the momentum of the marble is to increase its speed. The speed of the marble can be **increased** by throwing it using a **slingshot**, as shown in figure 7.2, thus, increasing its speed. When this stone hits the glass door, it breaks it.

### Expression and SI Unit of Momentum

Now, it can be concluded that the momentum is directly proportional to the moving object’s mass and speed or velocity. Therefore, the momentum can be mathematically written as,

**p = mv**

The SI unit of momentum can be derived by plugging the SI units of mass and speed in the formula above.

The SI unit of mass is ‘**kg**’ and of velocity is ‘**m/s**.’

Therefore, the SI unit of p = SI unit of mass x SI unit of velocity.

= kg * m/s

= **kg.m/s**

Thus, the SI unit of momentum is “**kg.m/s**“.

Momentum is a vector quantity, like velocity, in its formula, is a vector quantity. The momentum of an object is directed along with its velocity.

### The momentum of a body at rest:

A body at **rest** has **no** motion at all. For example, a tree is always at rest, i.e., it has no motion. Thus, momentum being the amount of motion contained in a body, is also **zero**.

Mathematically, if a body is at rest, then its speed is **zero**.

Thus, momentum = mass × speed

= mass × 0

= **0**

Thus, a body at rest has no momentum at all.

### Example 1:

An athlete weighing 64 kg runs a race at a speed of 12 m/s. When she gets closer to the finish line, she slows down to 8 m/s. What are her momenta in each case?

### Solution:

Mass of the athlete = 64 kg

Her initial speed = 12 m/s

Her final speed = 8 m/s

Therefore, her initial momentum = 64 × 12

= **768 kg.m/s**

And her final momentum = 64 × 8

= **512 kg.m/s**

### Example 2:

A motor car of mass 1100 kg is moving along a straight line with a uniform velocity of 90 km/h. Its velocity is slowed down to 18 km/h in 4 s by an unbalanced external force. Calculate the acceleration and the change in momentum.

### Solution:

Mass of the motor car = 1100 kg

Initial velocity = 90 km/h

Final velocity = 18 km/h

Time taken = 4 s

Initial velocity (u) = 90 km/h = 90 x (5/18) = 25 m/s

Final velocity (v) = 18 km/h = 18 x (5/18) = 5 m/s

Therefore, acceleration = (v – u)/t

= (5 – 25)/4

= 5 m/s^{2}

Initial momentum = 1100 x 25 = **27500 kg.m/s**

Final momentum = 1100 x 5 = **5500 kg.m/s**

### Example 3:

When a bullet is **thrown** on a person, it hardly hurts. However, when the same bullet is **fired** from a gun, it might kill a person or pierce them through. Why?

### Answer:

The bullet which is fired has a much larger velocity as compared to the one that is thrown. Thus, the former has a much greater momentum as compared to the latter. The bullet with an **increased velocity** and, thus, **increased momentum** can show a much greater **impact** as compared to the same bullet with **smaller momentum**.

### Example 4:

The bicycle, the truck, and the car are moving at the same speed of **5 m/s**. The car’s mass is **10** times that of the bicycle, whose mass is **10 kg** and the mass of the truck is **3 times** that of the car. Which one do you think is more fatal? Justify your answer.

### Solution:

Given that,

Mass of the bicycle = 10 kg

Mass of the car = 10 × mass of the bicycle

= 10 × 10 = 100 kg

Mass of the truck = 3 × mass of the car

= 3 × 100 = 300 kg

Velocity of all of them = 5 m/s

The fatalness of an accident depends directly on the momentum of the vehicle approaching the victim.

The momentum of the bicycle = 10 × 5

= **15 kg.m/s**

The momentum of the car = 100 × 5 = **500 kg.m/s**

The momentum of the truck = 300 × 5

= **1500 kg.m/s**.

The accident with the **truck** is the most fatal because the momentum of a truck approaching at 5 m/s is the highest.

### Summary:

- The amount of motion contained in a body is called momentum.
- Momentum is mathematically written as p = mv.
- It is a vector quantity, as velocity in its formula is a vector.
- The direction of the momentum vector is along with the velocity of the body.
- Its SI unit is kg.m/s.
- It depends directly upon the mass of the body and the velocity with which it is moving.
- The momentum possessed by a body at rest is zero, as it has zero velocity.
- The momentum of a body increases with an increase in its mass and/or velocity.

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