Table of Contents
Force, Momentum and Impulse
1. Introduction
Force, momentum and impulse are closely related concepts in mechanics.
Force describes an interaction that changes or tends to change the motion of an object. Momentum describes the quantity of motion associated with an object’s mass and velocity. Impulse measures the effect of a force acting over a period of time.
These concepts are essential for understanding collisions, vehicle safety, sports, recoil, rocket propulsion and the conservation of momentum.
2. Force
Force is an interaction capable of changing an object’s velocity, direction of motion or shape.
Force is a vector quantity.
The SI unit is the newton (N).
[
F=ma
]
This relationship applies to a constant-mass object in an inertial frame.
2.1 Effects of force
A force can:
- Set a stationary object in motion.
- Stop a moving object.
- Increase or decrease speed.
- Change the direction of motion.
- Change the shape or dimensions of an object.
- Produce rotational motion.
- Maintain circular motion by continually changing direction.
2.2 Contact and non-contact forces
Contact forces arise through physical contact between objects.
Examples include friction, tension, normal reaction, spring force and air resistance.
Non-contact forces act without direct physical contact.
Examples include gravitational, magnetic and electrostatic forces.
3. Types of Important Forces
3.1 Gravitational force
Gravitational force is the attractive force between objects possessing mass.
Near Earth’s surface, the gravitational force on an object is its weight.
[
W=mg
]
3.2 Frictional force
Friction is a force that opposes relative motion or the tendency of relative motion between surfaces in contact.
Types include:
- Static friction.
- Kinetic or sliding friction.
- Rolling resistance.
Static friction adjusts to oppose an applied force up to its limiting value. Kinetic friction acts when surfaces slide against one another.
Friction is useful in walking, vehicle braking and gripping objects, but it also causes energy loss and wear.
3.3 Normal reaction
The normal force is the contact force exerted by a surface perpendicular to that surface.
For an object resting on a horizontal surface with no other vertical forces, the normal force equals the object’s weight.
This equality does not apply in every situation, such as on inclined surfaces or in vertically accelerating systems.
3.4 Tension
Tension is the pulling force transmitted through a stretched string, rope or cable.
For an ideal massless rope passing over a frictionless pulley, tension is the same throughout the rope.
3.5 Air resistance
Air resistance is a resistive force exerted by air on a moving object.
It generally acts opposite to the object’s velocity relative to the air.
It depends on factors such as speed, shape, frontal area, air density and drag coefficient.
3.6 Elastic or spring force
An ideal spring exerts a restoring force proportional to its displacement from equilibrium, within its elastic range.
Hooke’s law is:
[
F=-kx
]
Where:
- (F) = restoring force
- (k) = spring constant
- (x) = displacement from equilibrium
The negative sign indicates that the spring force acts opposite to the displacement.
4. Resultant Force and Equilibrium
The resultant or net force is the vector sum of all forces acting on an object.
If forces act in the same direction, their magnitudes are added.
If forces act in opposite directions along the same straight line, their magnitudes are subtracted, taking direction into account.
4.1 Equilibrium
An object is in translational equilibrium when the net external force acting on it is zero.
[
\sum F=0
]
In equilibrium, the object has zero acceleration.
It may be:
- At rest, or
- Moving with constant velocity.
Rotational equilibrium additionally requires zero net torque.
5. Momentum
Momentum is the product of an object’s mass and velocity.
[
p=mv
]
Where:
- (p) = momentum
- (m) = mass
- (v) = velocity
The SI unit is kilogram metre per second (kg·m/s).
Momentum is a vector quantity because velocity is a vector.
5.1 Characteristics of momentum
- Momentum depends on mass and velocity.
- Greater mass produces greater momentum at the same velocity.
- Greater speed produces greater momentum for the same mass.
- Momentum has direction.
- Momentum can be positive, negative or zero in one-dimensional motion.
- A stationary object has zero linear momentum.
- Momentum is conserved in an isolated system.
5.2 Example: Calculating momentum
A vehicle has a mass of 1000 kg and moves at 20 m/s.
[
p=mv
]
[
p=1000\times20
]
[
p=20,000\ kg\cdot m/s
]
Its momentum is 20,000 kg·m/s in the direction of motion.
5.3 Momentum and kinetic energy
Momentum and kinetic energy are related but different physical quantities.
For a non-relativistic object:
[
KE=\frac{1}{2}mv^2
]
Momentum depends linearly on velocity, whereas kinetic energy depends on the square of velocity.
For a given mass, doubling velocity doubles momentum but quadruples kinetic energy.
6. Newton’s Second Law in Terms of Momentum
Newton’s second law states that the net external force is equal to the rate of change of momentum.
[
F_{\text{net}}=\frac{\Delta p}{\Delta t}
]
For an instantaneous rate of change:
[
F_{\text{net}}=\frac{dp}{dt}
]
If mass remains constant:
[
F_{\text{net}}=ma
]
This general form is especially useful when studying collisions and systems in which momentum changes rapidly.
7. Impulse
Impulse is the product of a force and the time interval during which it acts, when the force is constant.
[
J=F\Delta t
]
Where:
- (J) = impulse
- (F) = constant net force
- (\Delta t) = time interval
The SI unit is newton-second (N·s).
[
1\ N\cdot s=1\ kg\cdot m/s
]
Impulse is a vector quantity.
7.1 Impulse-momentum theorem
The impulse-momentum theorem states that the net impulse acting on an object equals its change in momentum.
[
J=\Delta p
]
Therefore:
[
J=p_f-p_i
]
For constant mass:
[
J=m(v-u)
]
This theorem is fundamental to understanding the effects of forces during collisions.
7.2 Example: Calculating impulse
A force of 50 N acts on an object for 4 seconds.
[
J=F\Delta t
]
[
J=50\times4=200\ N\cdot s
]
The impulse is 200 N·s.
7.3 Example: Change in momentum
A 2 kg ball changes its velocity from 3 m/s to 8 m/s in the same direction.
[
\Delta p=m(v-u)
]
[
\Delta p=2(8-3)
]
[
\Delta p=10\ kg\cdot m/s
]
The impulse is 10 N·s in the direction of motion.
8. Impulse and Collision Safety
The impulse-momentum theorem explains why increasing the time taken to stop a moving object can reduce the average force experienced.
For a fixed change in momentum:
[
F_{\text{avg}}=\frac{\Delta p}{\Delta t}
]
If the stopping time increases, the average force decreases.
8.1 Applications
Seat belts: They restrain occupants and increase the time over which their momentum is reduced.
Airbags: They provide a cushioning surface and increase the time over which a passenger is brought to rest.
Catching a ball: A player moves their hands backwards while catching a fast ball, increasing the stopping time and reducing the average force on the hands.
Sports equipment: Helmets, protective padding and safety mats help reduce impact forces.
Packaging: Cushioning materials increase the time over which a fragile object comes to rest during an impact.
9. Conservation of Momentum
The law of conservation of momentum states:
The total momentum of an isolated system remains constant if the net external force acting on the system is zero.
In a system containing two objects:
[
m_1u_1+m_2u_2=m_1v_1+m_2v_2
]
Where:
- (m_1,m_2) = masses of the objects
- (u_1,u_2) = initial velocities
- (v_1,v_2) = final velocities
The equation applies to a one-dimensional system. Direction must be represented using appropriate positive and negative signs.
9.1 Example: Two objects collide
A 2 kg object moving at 4 m/s collides with a stationary 2 kg object. They stick together after collision.
Calculate their common velocity, assuming no external impulse.
Initial momentum:
[
p_i=(2\times4)+(2\times0)=8
]
Total mass after collision:
[
2+2=4\ kg
]
Final velocity:
[
v=\frac{8}{4}=2\ m/s
]
The two objects move together at 2 m/s in the original direction.
10. Types of Collisions
A collision is an interaction between two or more bodies over a relatively short time, during which they exert significant forces on one another.
10.1 Elastic collision
In an elastic collision:
- Total momentum is conserved.
- Total kinetic energy is conserved.
Ideal elastic collisions are approximated in certain interactions, such as collisions between gas molecules under suitable conditions.
10.2 Inelastic collision
In an inelastic collision:
- Total momentum is conserved for an isolated system.
- Total kinetic energy is not conserved.
- Some kinetic energy is transformed into heat, sound, deformation or other forms of energy.
10.3 Perfectly inelastic collision
A perfectly inelastic collision occurs when colliding objects stick together and move with a common final velocity.
Momentum remains conserved in an isolated system, but the loss of kinetic energy is the maximum possible for the given total momentum and masses.
10.4 Comparison of collisions
| Type | Momentum conserved? | Kinetic energy conserved? | Objects stick together? |
|---|---|---|---|
| Elastic | Yes, in an isolated system | Yes | No, not necessarily |
| Inelastic | Yes, in an isolated system | No | Not necessarily |
| Perfectly inelastic | Yes, in an isolated system | No | Yes |
11. Recoil and Rocket Propulsion
11.1 Recoil
Recoil is the backward movement of an object when it expels or transfers momentum in the opposite direction.
For example, when a gun fires a bullet, the bullet moves forward and the gun experiences backward recoil.
The total momentum before and after firing remains conserved if the system is isolated.
11.2 Rocket propulsion
A rocket moves forward by expelling exhaust gases backwards at high speed.
The backward momentum of the exhaust gases is balanced by the forward momentum gained by the rocket, considering the complete system and external forces.
Rocket propulsion works in space because the rocket carries its own propellant and does not need air to push against.
12. Force, Momentum and Impulse: Comparison
| Concept | Meaning | SI unit | Nature |
|---|---|---|---|
| Force | Interaction that changes motion | N | Vector |
| Momentum | Mass multiplied by velocity | kg·m/s | Vector |
| Impulse | Change in momentum | N·s | Vector |
| Acceleration | Rate of change of velocity | m/s² | Vector |
Although impulse and momentum have different meanings, they share the same SI unit dimensions.
13. Practical Importance
Force, momentum and impulse have applications in:
- Vehicle crash safety.
- Sports and athletic performance.
- Rocket and spacecraft propulsion.
- Industrial machinery.
- Structural and earthquake engineering.
- Collision analysis and accident reconstruction.
- Packaging and material protection.
- Mechanical design and manufacturing.
Engineers use these principles to design safer vehicles, protective equipment, machines and structures.
36 Key Takeaways for Exams
- Force is an interaction capable of changing an object’s motion or shape.
- Force is a vector quantity.
- The SI unit of force is the newton.
- One newton equals one kg·m/s².
- Forces are classified as contact and non-contact forces.
- Friction opposes relative motion or its tendency between contacting surfaces.
- Static friction acts when surfaces do not slide relative to one another.
- Kinetic friction acts when surfaces slide relative to one another.
- Normal reaction acts perpendicular to a contact surface.
- Tension is a pulling force transmitted through a stretched rope or cable.
- Air resistance generally opposes motion relative to air.
- Hooke’s law describes the restoring force of an ideal spring within its elastic range.
- Net force is the vector sum of all forces acting on an object.
- Translational equilibrium requires zero net external force.
- Zero net force means zero acceleration, not necessarily rest.
- Momentum is the product of mass and velocity.
- Momentum is a vector quantity.
- The SI unit of momentum is kg·m/s.
- A stationary object has zero linear momentum.
- Doubling an object’s velocity doubles its momentum if mass remains constant.
- Newton’s second law relates net force to the rate of change of momentum.
- Impulse is the product of constant force and the time interval over which it acts.
- The SI unit of impulse is N·s.
- One N·s is equivalent to one kg·m/s.
- The impulse-momentum theorem states that impulse equals change in momentum.
- Increasing stopping time reduces average impact force for a fixed change in momentum.
- Seat belts and airbags help reduce injuries by controlling the stopping process.
- Catching a ball by moving the hands backwards increases stopping time.
- The law of conservation of momentum applies to an isolated system with zero net external force.
- Momentum is conserved in both elastic and inelastic collisions.
- Kinetic energy is conserved in an ideal elastic collision.
- Kinetic energy is not conserved in an inelastic collision.
- In a perfectly inelastic collision, objects stick together after impact.
- Recoil is explained by momentum conservation and Newton’s third law.
- Rockets propel themselves by expelling exhaust gases backwards.
- Force, momentum and impulse are essential concepts in collision analysis, vehicle safety and engineering design.