(a)  Measurement of force from Newton's second law. Suppose a force F acts on a body of mass m and changes its velocity from u to v in t seconds. Then
Initial momentum of the body, \({p}_{1}=mu\)
Final momentum of the body,  \({p}_{2}=mu\)
Change of momentum  \(= {p}_{2}-{p}_{1}=mv-mu=m\left(v-u\right)\)
Time taken \(= t\)
\(\therefore\) Rate of change of momentum  \(= \frac {Change \ of\ momentum} {time \ taken} = \frac {m\left(v-u\right)}{t}=ma\) 
Where a  is the acceleration of the body.
According to Newton's second law, the rate of change of momentum is directly proportional to the applied force, so
 \(F \infty \ ma.\)             or F = kma
Where k is constant. The unit of force is so chosen that k is equal to one. If m = 1, a = 1 and F = 1, then
1 = k. 1. 1    or  k = 1
F = ma     or   \(Force = Mass \times Acceleration.\)  
So a unit force is that force which produces a unit acceleration in a body of unit mass.
(b) The S.I. unit of force is Newton. One Newton is that force which produces an acceleration of \(1 m/ {s}^{2}\) in a body of mass 1
1 newton \(= 1 kg \times 1 m/s^2\) or \(1N = 1 kg m/s^2\)  
Thus, the second law of motion gives us a method to measure force. If the mass and acceleration of a body are known, we can determine the force acting on it.

Newton's first law of motion. According to this law, a body at rest or in uniform motion will remain at rest or in uniform motion unless an unbalanced force acts upon it. This law consists of three parts:
(i) First part, says that a body at rest continues in its state of rest. For example, a person standing in a bus falls backward when the bus suddenly starts moving forward. When the bus moves, the lower part of his body begins to move along with the bus while the upper part of his body continues to remain at rest due to inertia. That is why a person falls backward when the bus starts.
(ii) Second part, says that a body uniform motion continues moving in straight line path will a uniform speed. for example, when a moving bus suddenly stops, a person sittng in it falls forward. As the bus stops, the lower part of his body comes to rest along with the bus while upper part of his body continues to remain in motion due to inertia and so he fall forward.
(iii) Third part, says that a body moving with a uniform speed in a straight line cannot change itself its direction of motion. For example, when a bus takes a sharp turn, a person sitting in the bus gets a force acting away from the center of the curved path due to his tendency to move in the original direction. He has to hold on a support to prevent himself from swaying away in the turning bus.

In first case:         m = 50 kg,   F = 80 N
\(\therefore\) Acceleration,    \(a=\frac { F }{ m } =\frac { 80 }{ 60 } =1.6m/{ s }^{ 2 }\)
In second case:   m = 25 kg,    F = 80 N
\(\therefore\) Acceleration,    \(a=\frac { F }{ m } =\frac { 80 }{ 25 } =3.2m/{ s }^{ 2 }.\)

1. When a moving bus stops, the lower part of our body in contact with the bus comes to rest while the upper part of out body tends to keep moving due to inertia of motion. Hence, we fall downwards (or forwards).
2. When the bus accelerates from rest, the lower part of our body comes into motion along with the bus while the upper part of body tends to remain at rest due to inertia of rest. hence we fall backward.

A sudden movement of the vehicle results in the sudden change in the state of motion of the vehicle and of our feet in contact with it. But the rest of our body oppose this change due to its inertia and tends to remain where it was. Seat belts are provided to protect the passengers from falling backward or forward during such situations.

Inertia. Inertia is the natural tendency of a body to resist any change in its state of rest or uniform motion in a straight line. For example, a book lying on a table will remain there until an external force is applied on to remove or displace it from that position. Inertia os of three types:
(i) inertia of rest,
(ii) inertia of motion and
(iii) inertia of direction.

Inertia. Inertia is the natural tendency of a body to resist any change in its state of rest or uniform motion in a straight line. For example, a book lying on a table will remain there until an external force is applied on to remove or displace it from that position. Inertia os of three types:

(i) inertia of rest, (ii) inertia of motion and (iii) inertia of direction.

(a) Both vehicles experience equal forces of action and reaction.
(b) Momentum change for both vehicles is same.
(c) Car experiences greater acceleration due to its smaller mass.
(d) Due to its smaller inertia or opposite to the force exerted on it.

Here     m = 1,200 kg
Initial velocity,  \(v = 90 \ km/h = 90 \times \frac {5}{18} \ m \ s^{-1}=25 \ m s ^{-1}\)
Final velocity,    \(u = 18 km/h = 18 \times \frac {5}{18} = 5m\quad s^{-1}\)
Time,    t  =  4 s
Accleration,   \(a = \frac {v-u}{t}=\frac {5-25}{4}=-5 m \ s^{-2}\)
magnitude of accleration  \(= 5 m \ s^{-2}.\)
Change in momentum \(= m \left( v- u \right)=1,200 \left(5-25\right)\)
 \(= - 24,000 \ kq \ m \ s^{-1}.\)
Magnitude of change in momentum \(= 24,000 \ kq \ m \ s^{-1}.\)
Magnitude of force \(= \frac {Change \ in \ momentum}{time \ taken}=\frac {24,000}{4}=6,000 \ N.\)

Here,   \(m_1 = 1 \ kg,\)    \(u_2= 10 \ ms^{-1},\)   \(m_2 = 5 \ kg,\)   \(u_2=0\)
Let v be the velocity of the combined object after the collision
Total momentum just before the impact
 \(=m_1u_1+m_2u_2 = 1 \times 10 +5 \times 0 = 10\quad kg \quad ms^{-1}\)
Total momentum just after the impact
 \(= (m_1+m_2)v=(1+5)v=6 v \quad kg m s^{-1}\)
By conservation of momentum,
6v = 10
or    \(v = \frac {10}{6}=\frac {5}{3} m s^{-1}\)
\(\therefore\)   Total momentum just after the impact
 \(= 6 \times \frac {5}{3}=10 \quad m s^{-1}.\)

Mass of bullet, \(m_1=50 \ g = 0.05 kg\)
Mass of rifle,   \(m_2=4\) kg
Initial velocity of bullet, \(u_1=0\)
Initial velocity of rifle, \(u_2=0\)
Final velocity of bullet, \(v_1=35 ms^{-1}\)
Final velocity of rifle, \(v_2 = ?\)
According to the law of conservation of momentum,
total momenta after the fire = Total momenta before the fire
 \(m_1v_1+m_2v_2=m_1u_1+m_2u_2\)
 \(0.05 \times 35 + 4v_2=0+0\)
\({ v }_{ 2 }=-\frac { 0.05\times 35 }{ 4 } =-\frac { 7 }{ 16 } =-0.44{ ms }^{ -1 }\)
The negative sign indicates that the direction in which the rifle would recoil is opposite to that of the bullet.    

Law of conservation of momentum. This law states that if a number of bodies are interacting with each other (i.e., exerting forces on each other), their total momentum remains conserved before and after the interaction, provided there is no external force acting on them.
Derivation from Newton's second law of motion.  Let  \(p_1\)   and   \(p_2\)   represent the sum of momentum of a group of objects before and after the collision, respectively. Let t be the time elapsed during the collision.
According to Newton's Second law of motion,
 External force = Rate of change of momentum
or     \(F = \frac {p_2-p_1}{t}\)
If there is no external force, that is F = 0, then
\(\frac {p_2-{p}_{1}}{t}=0\) or \(p_2 = p_1\)   
Hence in the absence of an external force, the total momentum of a group of objects remains unchanged or conserved during the collision. This is the law of conservation of momentum.