AS/A Level Physics Momentum and Newton’s laws of motion Study Notes -2025-2027 Syllabus
AS/A Level Physics Momentum and Newton’s laws of motion Study Notes
AS/A Level Physics Momentum and Newton’s laws of motion Study Notes at IITian Academy focus on specific topic and type of questions asked in actual exam. Study Notes focus on AS/A Level Physics Study Notes syllabus with Candidates should be able to:
- understand that mass is the property of an object that resists change in motion
- recall F = ma and solve problems using it, understanding that acceleration and resultant force are always in the same direction
- define and use linear momentum as the product of mass and velocity
- define and use force as rate of change of momentum
- state and apply each of Newton’s laws of motion
- describe and use the concept of weight as the effect of a gravitational field on a mass and recall that the weight of an object is equal to the product of its mass and the acceleration of free fall
Mass as the Property that Resists Change in Motion
Mass is a physical property of matter that quantifies both the amount of substance in an object and its resistance to changes in motion. This resistance to acceleration when a resultant force acts is known as inertia.
Concept of Inertia:
- Every object tends to resist changes in its state of motion (whether at rest or moving).
- This resistance is directly related to the object’s mass.
- The greater the mass of an object, the greater the force required to change its motion (to start, stop, or change direction).
Key Idea:
Mass measures the degree of inertia of a body. An object with larger mass offers greater resistance to acceleration under the same applied force.
\( \mathrm{a = \dfrac{F}{m}} \ \Rightarrow \text{for a given } F, \text{ larger } m \Rightarrow \text{ smaller } a. \)
Unit:
The SI unit of mass is the kilogram (kg). Mass is a scalar quantity and remains the same everywhere in the universe.
Relation to Newton’s First Law:
According to Newton’s First Law, an object will remain at rest or move with constant velocity unless acted upon by a resultant force. Mass quantifies the object’s resistance to this change — this is why it is often described as the “measure of inertia.”
Summary Points:
- Mass is a measure of the amount of matter and the inertia of a body.
- Greater mass → greater resistance to change in motion.
- Mass does not change with location.
- Force is required to change the velocity of an object, and the magnitude of this force depends on its mass.
Example
A car of mass \( \mathrm{1000\,kg} \) and a truck of mass \( \mathrm{4000\,kg} \) are both initially at rest. The same engine force of \( \mathrm{2000\,N} \) acts on each. Which vehicle will accelerate more quickly?
▶️ Answer / Explanation
Using \( \mathrm{a = F/m} \):
Car: \( \mathrm{a = 2000/1000 = 2.0\,m/s^2} \)
Truck: \( \mathrm{a = 2000/4000 = 0.5\,m/s^2} \)
The truck’s larger mass gives it greater inertia, so it accelerates less under the same force.
Example
A satellite of mass \( \mathrm{1000\,kg} \) moving in space at constant velocity \( \mathrm{7500\,m/s} \) continues to move indefinitely even after its engines are turned off.
▶️ Explanation
In space, there is negligible air resistance or external force. The satellite’s mass provides inertia, resisting any change in its motion. Hence, it keeps moving at the same velocity, demonstrating that mass resists acceleration or deceleration.
Newton’s Second Law of Motion: ( \mathrm{F = ma} )
Newton’s Second Law states that the rate of change of momentum of an object is directly proportional to the resultant force acting on it, and this change occurs in the same direction as the force.
For constant mass, this law becomes:
\( \mathrm{F = ma} \)
This means that when a net (unbalanced) force acts on a body, it produces an acceleration proportional to the force and inversely proportional to the object’s mass.
Mathematical Form:
\( \mathrm{F = ma} \)
where:
- \( \mathrm{F} \): resultant force (N)
- \( \mathrm{m} \): mass (kg)
- \( \mathrm{a} \): acceleration (m/s²)
Direction of Force and Acceleration:
- The acceleration of an object always acts in the same direction as the resultant force.
- If multiple forces act on a body, the resultant (vector sum) determines the direction and magnitude of acceleration.
Physical Meaning:
Force is the cause of acceleration. The greater the resultant force on a given mass, the greater the acceleration produced. For a constant applied force, increasing the mass reduces the acceleration.
\( \mathrm{a = \dfrac{F}{m}} \)
Unit of Force:
One newton (N) is the force required to accelerate a mass of one kilogram by one metre per second squared.
\( \mathrm{1\,N = 1\,kg\,m/s^2} \)
Key Relationships:
- For a given mass: \( \mathrm{a \propto F} \)
- For a given force: \( \mathrm{a \propto \dfrac{1}{m}} \)
- If \( \mathrm{F = 0} \), then \( \mathrm{a = 0} \) → the body remains at rest or moves at constant velocity.
Summary Points:
- The resultant force on a body equals its mass multiplied by its acceleration.
- Force and acceleration are vector quantities and act in the same direction.
- This law connects Newton’s First Law (when \( \mathrm{F = 0} \)) and Third Law (action–reaction).
Example
A car of mass \( \mathrm{1000\,kg} \) experiences a resultant forward force of \( \mathrm{2000\,N.} \) Find the acceleration of the car.
▶️ Answer / Explanation
Using \( \mathrm{F = ma} \):
\( \mathrm{a = \dfrac{F}{m} = \dfrac{2000}{1000} = 2.0\,m/s^2.} \)
The car accelerates at \( \mathrm{2.0\,m/s^2} \) in the direction of the force.
Example
A stone of mass \( \mathrm{0.5\,kg} \) falls freely under gravity. Calculate the resultant force acting on it, taking \( \mathrm{g = 9.8\,m/s^2.} \)
▶️ Answer / Explanation
Using \( \mathrm{F = ma} \):
\( \mathrm{F = (0.5)(9.8) = 4.9\,N.} \)
The downward resultant force on the stone is \( \mathrm{4.9\,N.} \)
Linear Momentum
Linear momentum is defined as the product of an object’s mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction (the same direction as the object’s velocity).
\( \mathrm{\vec{p} = m\vec{v}} \)
Explanation:
- Momentum quantifies how difficult it is to stop or change the motion of a moving object.
- For a given velocity, a more massive object has greater momentum.
- Momentum depends on both the mass and velocity of the body.
Units:
SI unit: \( \mathrm{kg\,m/s} \)
Key Idea:
An object’s momentum changes when a resultant force acts on it. The greater the momentum, the more force or time is required to bring it to rest or change its direction.
Conservation of Linear Momentum:
In an isolated system (no external resultant force), the total linear momentum remains constant. This principle is the basis for collision and explosion analysis in mechanics.
Summary Points:
- \( \mathrm{\vec{p} = m\vec{v}} \)
- Momentum is a vector quantity.
- SI unit: \( \mathrm{kg\,m/s} \)
- Total momentum of an isolated system is conserved.
Example
A car of mass \( \mathrm{1200\,kg} \) is moving at \( \mathrm{25\,m/s.} \) Find its linear momentum.
▶️ Answer / Explanation
\( \mathrm{p = mv = 1200 \times 25 = 3.0 \times 10^4\,kg\,m/s.} \)
The car’s momentum is \( \mathrm{3.0 \times 10^4\,kg\,m/s} \) in the direction of motion.
Example
A 2 kg ball moving at \( \mathrm{10\,m/s} \) and a 4 kg ball moving at \( \mathrm{5\,m/s} \) have the same momentum. Verify this statement.
▶️ Answer / Explanation
For 2 kg ball: \( \mathrm{p_1 = 2 \times 10 = 20\,kg\,m/s.} \)
For 4 kg ball: \( \mathrm{p_2 = 4 \times 5 = 20\,kg\,m/s.} \)
Hence, both have equal momenta of \( \mathrm{20\,kg\,m/s.} \)
Force as the Rate of Change of Momentum
Force is defined as the rate of change of momentum of an object. This is the general form of Newton’s Second Law, applicable even when the mass is not constant.
\( \mathrm{\vec{F} = \dfrac{d\vec{p}}{dt}} \)
Explanation:
- If the momentum of an object changes (either its speed, direction, or both), a net force must have acted on it.
- The direction of the resultant force is the same as the direction of change of momentum.
- This definition is consistent with \( \mathrm{F = ma} \) when mass is constant, since:
\( \mathrm{F = \dfrac{d(mv)}{dt} = m\dfrac{dv}{dt} = ma.} \)
Units:
SI unit: newton (N) \( \mathrm{1\,N = 1\,kg\,m/s^2} \)
Physical Meaning:
- A force causes a change in the momentum of a body.
- The greater the rate of change of momentum, the greater the force required.
- If momentum changes slowly, the average force is small; if it changes rapidly, the average force is large.
Impulse Relationship:
When a force acts for a short time \( \mathrm{t} \), the product of force and time gives the change in momentum:
\( \mathrm{F \Delta t = \Delta p} \)
This product \( \mathrm{F \Delta t} \) is called impulse.
Summary Points:
- \( \mathrm{\vec{F} = \dfrac{d\vec{p}}{dt}} \)
- Force is directly proportional to the rate of change of momentum.
- For constant mass, this simplifies to \( \mathrm{F = ma.} \)
- The direction of the resultant force equals the direction of momentum change.
Example
A tennis ball of mass \( \mathrm{0.060\,kg} \) changes velocity from \( \mathrm{10\,m/s} \) to \( \mathrm{-15\,m/s} \) (opposite direction) in \( \mathrm{0.020\,s.} \) Find the average force acting on it.
▶️ Answer / Explanation
Change in momentum: \( \mathrm{\Delta p = m(v – u) = 0.060(-15 – 10) = -1.5\,kg\,m/s.} \)
Average force: \( \mathrm{F = \dfrac{\Delta p}{\Delta t} = \dfrac{-1.5}{0.020} = -75\,N.} \)
The negative sign indicates the force acts opposite to the initial motion.
Example
A rocket ejects exhaust gases at high speed. The momentum of the gases changes rapidly, so an equal and opposite force (thrust) acts on the rocket.
▶️ Explanation
The rocket’s thrust can be expressed using \( \mathrm{F = \dfrac{d(mv)}{dt}} \), accounting for both changing mass and velocity. This is the general form of Newton’s Second Law — it explains how rockets accelerate even in space, where there is no external medium.
Newton’s Laws of Motion
Newton’s three laws of motion describe the relationship between forces acting on a body and the motion of that body. Together, they form the foundation of classical mechanics.
Newton’s First Law — Law of Inertia
Statement:
An object will remain at rest, or continue to move in a straight line at constant velocity, unless acted upon by a resultant external force.
Explanation:
- This law defines inertia — the tendency of an object to resist changes in its state of motion.
- If the resultant force on a body is zero, then:
- If at rest → it stays at rest.
- If moving → it continues with constant velocity in a straight line.
Example
A spacecraft is moving at a constant velocity in deep space with its engines switched off. Explain, using Newton’s First Law, why it continues to move at the same velocity.
▶️ Answer / Explanation
In deep space, there are negligible external forces such as friction or air resistance. According to Newton’s First Law, an object continues to move with constant velocity if no resultant external force acts on it. Therefore, the spacecraft maintains its velocity due to its inertia.
Newton’s Second Law — Relation Between Force and Acceleration
Statement:
The rate of change of momentum of a body is directly proportional to the resultant force acting on it and takes place in the direction of that force.
For constant mass:
\( \mathrm{\vec{F} = m\vec{a}} \)
Explanation:
- The acceleration produced in an object is directly proportional to the resultant force and inversely proportional to its mass.
- If \( \mathrm{F = 0} \), then \( \mathrm{a = 0} \) — the object remains at rest or moves with constant velocity.
Example
A car of mass \( \mathrm{800\,kg} \) experiences a resultant forward force of \( \mathrm{1600\,N.} \) Calculate its acceleration and state the direction of motion.
▶️ Answer / Explanation
Using \( \mathrm{F = ma} \):
\( \mathrm{a = \dfrac{F}{m} = \dfrac{1600}{800} = 2.0\,m/s^2.} \)
The car accelerates at \( \mathrm{2.0\,m/s^2} \) in the direction of the resultant (forward) force.
Newton’s Third Law — Action and Reaction
Statement:
For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts an equal and opposite force on object A.
Explanation:
- The two forces are equal in magnitude, opposite in direction, and act on different bodies.
- They do not cancel out because they act on separate objects.
- Action–reaction pairs always occur simultaneously.
Example
A book rests on a table. Describe the action and reaction forces between the book and the table.
▶️ Answer / Explanation
The book exerts a downward force on the table equal to its weight (action). The table exerts an equal and opposite upward normal reaction force on the book (reaction). These forces are equal in magnitude, opposite in direction, and act on different objects.
Weight and the Gravitational Field
Definition of Weight:
Weight is the force experienced by a mass due to the gravitational field of a planet or celestial body. It is the product of the object’s mass and the local acceleration due to gravity.
\( \mathrm{W = mg} \)
Explanation:
- Weight is a force and therefore a vector quantity — it acts vertically downward towards the centre of the Earth (or other attracting body).
- Mass is constant, but weight varies with the local value of \( \mathrm{g} \).
- On the Moon, where gravity is weaker, an object’s weight is less, though its mass is unchanged.
Units:
- Weight: newton (N)
- Mass: kilogram (kg)
- Acceleration due to gravity: \( \mathrm{m/s^2} \)
Concept of Gravitational Field:
A gravitational field is a region in which a mass experiences a force of attraction due to another mass. The field strength \( \mathrm{g} \) at any point is defined as the force per unit mass acting on a small test mass placed at that point:
\( \mathrm{g = \dfrac{F}{m}} \)
Hence, rearranging gives the definition of weight:
\( \mathrm{W = mg} \)
Key Characteristics:
- Weight depends on the strength of the gravitational field.
- Weight acts vertically downwards.
- Weight and mass are related by the constant \( \mathrm{g} \) (≈ \( \mathrm{9.81\,m/s^2} \) near the Earth’s surface).
Summary Points:
- Weight is the gravitational force on a mass: \( \mathrm{W = mg.} \)
- Weight varies with gravitational field strength; mass remains constant.
- Gravitational field strength is defined as \( \mathrm{g = \dfrac{F}{m}}. \)
- Weight always acts vertically downwards towards the Earth’s centre.
Example
Find the weight of a student of mass \( \mathrm{65\,kg} \) on Earth, where \( \mathrm{g = 9.81\,m/s^2.} \)
▶️ Answer / Explanation
\( \mathrm{W = mg = 65 \times 9.81 = 637.7\,N.} \)
The student’s weight on Earth is approximately \( \mathrm{638\,N.} \)
Example
For the same student on the Moon (\( \mathrm{g = 1.63\,m/s^2} \)), determine the weight.
▶️ Answer / Explanation
\( \mathrm{W = 65 \times 1.63 = 106.0\,N.} \)
The student’s weight on the Moon is much smaller due to the weaker gravitational field, though their mass remains \( \mathrm{65\,kg.} \)