Unit 3.3 – Momentum

What is Momentum?

Momentum is a measurement of the motion of an object. It depends on two factors:

1. Mass of the object ($m$).
2. Velocity of the object ($v$).

Mathematically, momentum ($p$) is given by:

$$p=m\cdot v$$

• Units: The SI unit of momentum is $\text{kg}\text{cdotp}\text{m/s}$.
• Nature: Momentum is a vector quantity, meaning it has both magnitude and direction (same as velocity).

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Key Points About Momentum

1. Dependence on Mass and Velocity:
   • A heavier object moving at the same speed as a lighter object will have more momentum.
     Example: A truck moving at $10\text{m/s}$ has more momentum than a bicycle moving at the same speed.
   • A faster-moving object will have more momentum than a slower-moving one of the same mass.
     Example: A bullet fired from a gun has much higher momentum than the same bullet rolling on the ground.
2. Direction of Momentum:
   • Momentum always points in the same direction as the velocity.
     Example: If a car moves eastward, its momentum is also directed eastward.
3. Conservation of Momentum:
   • In a closed system (no external forces), the total momentum before and after an event remains constant. This is called the Law of Conservation of Momentum.
     Example: During a collision between two cars, the total momentum of the system before and after the collision remains the same.

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Applications of Momentum

1. Collisions:
   • Momentum helps us analyze what happens during collisions.
   • Example: When two billiard balls collide, their total momentum before and after the collision remains the same if no external force acts on them.
2. Sports:
   • Momentum plays a key role in sports like football, cricket, and hockey.
   • Example: A fast-moving football has high momentum, making it harder to stop.
3. Rocket Propulsion:
   • Rockets expel gases downward with high momentum, and the reaction force propels the rocket upward.
4. Safety Measures:
   • Airbags in cars reduce injuries by increasing the time over which momentum changes, thereby reducing the force experienced by passengers.

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Relation Between Momentum and Kinetic Energy

The kinetic energy ($KE$) of an object is related to its momentum ($p$) by the formula:

$$KE=\frac{p\cdot p}{2m}$$

• If momentum doubles, kinetic energy becomes four times greater.
• Example: Doubling the speed of a car doubles its momentum but increases its kinetic energy by four times.

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Examples to Understand Momentum

1. Example 1:
   • A car of mass $1000\text{kg}$ is moving at $20\text{m/s}$. What is its momentum?
   • Solution:
     $$p=m\cdot v=1000\cdot 20=20,000\text{kg}\text{cdotp}\text{m/s}\mathrm{.}$$
2. Example 2:
   • A bullet of mass $0.01\text{kg}$ is fired with a velocity of $500\text{m/s}$. What is its momentum?
   • Solution:
     $$p=m\cdot v=0.01\cdot 500=5\text{kg}\text{cdotp}\text{m/s}\mathrm{.}$$
3. Example 3:
   • A truck of mass $5000\text{kg}$ is moving at $10\text{m/s}$. A car of mass $1000\text{kg}$ is moving at $50\text{m/s}$. Which has more momentum?
   • Solution:
     • Truck:
     $$p_{\text{truck}}=5000\cdot 10=50,000\text{kg}\text{cdotp}\text{m/s}\mathrm{.}$$
     • Car:
     $$p_{\text{car}}=1000\cdot 50=50,000\text{kg}\text{cdotp}\text{m/s}\mathrm{.}$$
     • Both have the same momentum.

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Impulse and Change in Momentum

Impulse is the change in momentum caused by a force acting over a period of time.

$$\text{Impulse}=F\cdot t=\mathrm{\Delta }p$$

Where:

• $F$ = Force applied.
• $t$ = Time for which the force acts.
• $\mathrm{\Delta }p$ = Change in momentum.

Real-Life Examples of Impulse:

1. Catching a Ball:
   • A cricket player lowers their hands while catching a ball to increase the time of contact, reducing the force exerted on their hands.
2. Car Crashes:
   • Airbags increase the time over which the momentum of passengers changes, reducing the force they experience.
3. Hitting a Nail:
   • A hammer delivers a large force over a short time, causing a significant change in momentum to drive the nail into the wood.

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Practice Questions

1. What is the momentum of a $5\text{kg}$ object moving at $8\text{m/s}$?
   • Answer:
   $$p=m\cdot v=5\cdot 8=40\text{kg}\text{cdotp}\text{m/s}\mathrm{.}$$
2. A car of mass $1200\text{kg}$ accelerates from $10\text{m/s}$ to $20\text{m/s}$. What is the change in momentum?
   • Answer:
   $$\mathrm{\Delta }p=m\cdot \mathrm{\Delta }v=1200\cdot (20-10)=1200\cdot 10=12,000\text{kg}\text{cdotp}\text{m/s}\mathrm{.}$$
3. A force of $50\text{N}$ acts on an object for $4\text{s}$. What is the impulse delivered?
   • Answer:
   $$\text{Impulse}=F\cdot t=50\cdot 4=200\text{N}\text{cdotp}\text{s}\mathrm{.}$$
4. If the momentum of an object is doubled, what happens to its kinetic energy?
   • Answer:
     Kinetic energy becomes four times greater because $KE\propto p\cdot p$.
5. Why do airbags reduce injuries during car crashes?
   • Answer:
     Airbags increase the time over which momentum changes, reducing the force experienced by passengers.

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Quick Recap

1. Momentum: $p=m\cdot v$; vector quantity; SI unit = $\text{kg}\text{cdotp}\text{m/s}$.
2. Conservation of Momentum: Total momentum in a closed system remains constant.
3. Impulse: Change in momentum; $\text{Impulse}=F\cdot t=\mathrm{\Delta }p$.
4. Applications: Collisions, sports, rocket propulsion, safety measures.