Momentum and Newton's Laws: Principles, Calculations, and Safety

Definition and Formula: Momentum is a vector quantity possessed by masses in motion. It is defined as the product of an object's mass and its velocity.
- The equation for momentum is: $\text{momentum} = \text{mass} \times \text{velocity}$
- Represented symbolically as: $p = m \times v$
Units of Measurement: Momentum is measured in kilogram metres per second ( $kg\,m/s$ ), provided that mass ( $m$ ) is in kilograms ( $kg$ ) and velocity ( $v$ ) is in metres per second ( $m/s$ ).
Conceptual Understanding:
- Momentum is a measure of how difficult it is to stop a moving object.
- It increases proportionally with both the mass and the speed (velocity) of the object.
- Distinction: Momentum is a property of moving masses and should not be confused with "moment," which refers to the turning effect of a force.

Newton's Discovery: Sir Isaac Newton ( $1642-1727$ ) identified that when an unbalanced force acts on an object, it causes a change in momentum in the direction of that force.
Newton's Second Law (Precise Statement): The rate of change of momentum of an object is proportional to the force applied to that object. If the force is doubled, the momentum changes twice as quickly.
The Force and Momentum Relationship Equation: $\text{force, } F = \frac{\text{change in momentum}}{\text{time taken}}$ $F = \frac{mv - mu}{t}$
- Where $mu$ is initial momentum, $mv$ is final momentum, and $t$ is the time over which the force acts.
Derivation of $F = ma$ :
- The equation can be rearranged assuming mass remains constant: $F = \frac{m(v - u)}{t}$
- Since acceleration ( $a$ ) is defined as $\frac{v - u}{t}$ , the equation becomes: $F = m \times a$
Real-World Considerations:
- In situations like a space shuttle launch, the mass is not constant because fuel is burned and rocket stages are jettisoned.
- External factors like air resistance must also be factored in by rocket scientists.

The Law of Conservation of Momentum: In any system, total momentum is always conserved provided no external forces act on the system.
- $\text{Total momentum before collision} = \text{Total momentum after collision}$
Newton's Cradle: This device demonstrates the conservation law. When one ball is released and collides with the rest, one ball at the opposite end swings away with the same momentum.
Impulse: The product of force and time ( $F \times t$ ) is referred to as "impulse."
- A larger force applied for a longer time results in a greater change in momentum.
Collisions and Newton's Third Law: During a collision between two objects (e.g., Ball A and Ball B), each exerts an equal and opposite force on the other for the same amount of time ( $F \times t$ ).
- The increase in momentum of Ball B is exactly balanced by the decrease in momentum of Ball A.
Friction as an External Force: In a snooker game, friction from the table eventually stops the balls, reducing their momentum to zero. However, this momentum is not "lost" to the universe; rather, the table and Earth gain an equal amount of momentum (though unnoticeable due to their large mass).

Definition of Explosion: An explosion involves a release of energy that causes objects to fly apart. Momentum remains unchanged (conserved), though kinetic (movement) energy increases significantly.
Rocket Propulsion: Rockets work in the vacuum of space based on the principle of conservation of momentum.
- The rocket motor forces fast-moving gases (produced by burning fuel) out of the back of the rocket.
- The spacecraft gains an equal amount of momentum in the opposite direction of the moving exhaust gases.
Balloon Example: Releasing an inflated balloon without tying the end mimics this effect, as air escaping in one direction propels the balloon in the opposite direction.

Example 1: Rocket Momentum
- Scenario: A Moon mission rocket provides an unbalanced upward force of $30\,MN$ and burns for $2.5\,\text{minutes}$ . Total mass is $3000\,\text{tonnes}$ .
- Conversion: $1\,MN = 10^{6}\,N$ ; $1\,\text{tonne} = 1000\,kg$ .
- Force ( $F$ ) = $3 \times 10^{7}\,N$ .
- Time ( $t$ ) = $2.5 \times 60 = 150\,s$ .
- a) Increase in momentum: $F \times t = (3 \times 10^{7}\,N) \times (150\,s) = 4.5 \times 10^{9}\,kg\,m/s$ .
- b) Velocity ( $v$ ) if starting from rest ( $u=0$ ): $v = \frac{4.5 \times 10^{9}\,kg\,m/s}{3 \times 10^{6}\,kg} = 1.5 \times 10^{3}\,m/s$
Example 2: Railway Truck Collision
- Scenario: A $5000\,kg$ truck moving at $3\,m/s$ collides with a stationary $10,000\,kg$ truck and they join together.
- Conservation equation: $(m_{1} \times u) + (m_{2} \times 0) = (m_{1} + m_{2}) \times v$
- Calculation: $(5000 \times 3) + 0 = (15,000) \times v$
- Result: $v = \frac{15,000}{15,000} = 1\,m/s$ in the same direction as the original truck.

Force Reduction Principle: Safety features are designed to increase the time taken for momentum to change during an accident, thereby reducing the impact force.
The Equation Context: From $F = \frac{\Delta p}{t}$ , increasing time ( $t$ ) decreases force ( $F$ ) for the same change in momentum ( $\Delta p$ ).
Safety Features:
- Crumple Zones: Areas in the front and back of cars designed to collapse during a collision, increasing the deceleration time.
- Air Bags: Inflate rapidly to provide a soft surface that increases the time over which the passenger's head comes to rest.
- Seat Belts: Ensure the passenger decelerates with the car's crumple zones rather than continuing forward and hitting a hard interior surface.
- Escape Lanes: Filled with deep, soft sand to slow heavy lorries slowly, minimizing injury to the driver.
Example 3: Dash Impact vs. Safety Features
- A woman ( $50\,kg$ ) travelling at $20\,m/s$ hits a hard dashboard and stops in $0.02\,s$ .
- Force Calculation: $\frac{(50 \times 20) - (50 \times 0)}{0.02} = 50,000\,N$ .
- If the time is increased to $1\,s$ via safety features, the force drops to $1000\,N$ .

Newton's First Law: Objects remain at a steady speed in a straight line or remain stationary unless acted upon by a resultant (unbalanced) force.
Newton's Second Law: Acceleration is proportional to force and inversely proportional to mass ( $F = ma$ ).
Newton's Third Law: For every action, there is an equal and opposite reaction.
- Action/Reaction vs. Balanced Forces:
  - Balanced forces act in opposite directions on the same object.
  - Action and reaction forces act in opposite directions but on different objects (e.g., your weight pushes down on a seat, and the seat pushes back up on you).

Question 1: Basic Momentum Calculation
- a) Bowling ball: $6\,kg$ at $8\,m/s$ .
- b) Ship: $50,000\,kg$ at $3\,m/s$ .
- c) Tennis ball: $60\,g$ at $180\,km/h$ (Requires conversion of grams to kg and km/h to m/s).
Question 2: Air Rifle Pellet Impact
- Pellet ( $2\,g$ ) fired into a truck and clay ( $0.1\,kg$ ). Truck moves at $0.8\,m/s$ .
- a) Calculate momentum of truck/clay after collision.
- b) State pellet momentum before collision (conserved from part a).
- c) Calculate initial pellet velocity.
- d) State assumptions (e.g., negligible friction/air resistance).
Question 3: Rocket Impulse
- Rocket ( $1200\,kg$ ) at $2000\,m/s$ . Thrust = $10\,kN$ for $1\,\text{minute}$ .
- a) Calculate increase in momentum.
- b) Calculate new velocity after firing.
Question 4: Air Bags
- a) Discussion on whether air bags remain inflated (No, they deflate to absorb energy).
- b) Explanation of how increasing the impact time reduces force on the driver.