Physics Study Notes: Momentum (1D)
Hello! Welcome to the world of momentum. Ever wondered how a tiny, fast-moving bullet can have a huge impact, or how airbags in cars save lives? The answer lies in the concept of momentum. In these notes, we'll break down everything you need to know about momentum, impulse, and collisions. Don't worry if it sounds complicated; we'll use simple examples to make it clear. Let's get moving!
1. What is Momentum? It's Mass in Motion!
Imagine a bowling ball and a tennis ball rolling towards you at the same, slow speed. Which one would you rather stop? The tennis ball, of course! Why? Because the bowling ball has more mass. Now, imagine two tennis balls. One is thrown gently, and the other is fired from a cannon. Which is harder to stop? The one from the cannon, because it has a much higher velocity.
This idea of how 'hard an object is to stop' is what physicists call linear momentum. It’s a measure of an object's 'quantity of motion', combining both its mass and its velocity.
Defining Linear Momentum (p)
Momentum is a product of an object's mass and its velocity. We use the letter p to represent momentum.
The formula is simple:
$$ p = m \times v $$Where:
p = momentum (in kg m s⁻¹)
m = mass (in kg)
v = velocity (in m s⁻¹)
Momentum is a Vector!
This is SUPER important! Because velocity has a direction, momentum also has a direction. In 1D problems, we show this with positive (+) and negative (-) signs.
For example, if we decide 'right' is the positive direction:
- A car moving to the right has a positive (+) momentum.
- A car moving to the left has a negative (-) momentum.
Common Mistake Alert: Forgetting to include the correct sign for velocity is the #1 reason students make mistakes in momentum problems. Always define your positive direction first!
Units of Momentum
The standard unit is kilogram-metres per second (kg m s⁻¹). You might also see it written as newton-seconds (N s). They are exactly the same thing!
Quick Review
What is Momentum? A measure of an object's mass in motion.
Formula: $$p = mv$$
Key Property: It's a vector! Direction matters.
Key Takeaway: Momentum tells you how much 'oomph' a moving object has. More mass or more velocity means more momentum.
2. Change in Momentum & Impulse
To change an object's velocity (to accelerate it), you need to apply a net force. Since momentum depends on velocity, it makes sense that a net force is needed to change an object's momentum. This is the core of Newton's Second Law!
Newton's Second Law (The Momentum Version)
You probably know Newton's Second Law as $$F = ma$$. But there's a more fundamental way to state it:
The net force acting on an object is equal to the rate of change of its momentum.
In formula terms:
$$ F_{net} = \frac{\Delta p}{\Delta t} = \frac{p_{final} - p_{initial}}{\Delta t} = \frac{mv - mu}{\Delta t} $$Where:
$$F_{net}$$ = Net Force (N)
$$\Delta p$$ = Change in momentum (kg m s⁻¹)
$$\Delta t$$ = Time interval over which the force acts (s)
Introducing Impulse (J)
If we rearrange the formula above, we get something very useful:
$$ F_{net} \times \Delta t = \Delta p $$This quantity, $$F_{net} \times \Delta t$$, is called impulse. So, impulse is simply equal to the change in momentum.
Impulse = Change in Momentum
This gives us a powerful real-world connection. To produce a certain change in momentum ($$\Delta p$$), you can either use a BIG force for a short time, or a small force for a LONG time.
Real World Examples:
- Airbags & Crumple Zones: When a car crashes, the change in momentum ($$\Delta p$$) is fixed (from high speed to zero). Airbags and crumple zones increase the time ($$\Delta t$$) of the impact. Since $$F = \Delta p / \Delta t$$, increasing $$\Delta t$$ drastically decreases the force (F) on the passengers, saving lives.
- Bending your knees: When you jump down from a height, you bend your knees upon landing. This increases the time it takes to stop, reducing the force on your legs.
- Follow-through in sports: When hitting a tennis ball or golf ball, players 'follow through' with their swing. This keeps the racquet/club in contact with the ball for longer ($$\Delta t$$), delivering a larger impulse ($$F \Delta t$$) and therefore a greater change in momentum ($$\Delta p$$), making the ball go faster!
Key Takeaway: Impulse is the 'kick' you give to an object. It's the force applied multiplied by the time it's applied for, and it equals the object's change in momentum. To reduce the sting of an impact, increase the impact time!
3. The Law of Conservation of Momentum
This is one of the most important laws in all of physics! It's simple but incredibly powerful.
The Law of Conservation of Linear Momentum states that for a closed system (one with no external net forces like friction), the total momentum before a collision or interaction is equal to the total momentum after the interaction.
In simpler words: Momentum isn't lost, it's just transferred.
For a collision between two objects (object 1 and object 2), the formula is:
$$ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 $$Where:
m₁ , m₂ = masses of the objects
u₁ , u₂ = initial velocities (before collision)
v₁ , v₂ = final velocities (after collision)
How does this relate to Newton's Third Law?
Great question! Newton's Third Law says that for every action, there is an equal and opposite reaction. During a collision:
- Force on object 1 by object 2 (let's call it F₁₂) is equal and opposite to the force on object 2 by object 1 (F₂₁). So, $$F_{12} = -F_{21}$$
- The time of contact ($$\Delta t$$) is the same for both.
- Therefore, the impulse on object 1 ($$F_{12} \Delta t$$) is equal and opposite to the impulse on object 2 ($$-F_{21} \Delta t$$).
- Since impulse equals change in momentum, this means $$ \Delta p_1 = - \Delta p_2 $$.
- Rearranging this gives $$ \Delta p_1 + \Delta p_2 = 0 $$. The total change in momentum for the whole system is zero. If the total momentum doesn't change, it must be conserved!
Did you know?
This law explains how rockets work in the vacuum of space. There's no air to push against. Instead, the rocket throws hot gas out of its engine at high speed. The gas gets momentum in one direction, so to conserve the total momentum (which was initially zero), the rocket must get an equal amount of momentum in the opposite direction!
Key Takeaway: In any collision or explosion where no external forces are involved, the total momentum of all objects added together stays exactly the same.
4. Types of Collisions (in 1D)
While momentum is always conserved in a closed system, the same isn't always true for kinetic energy ($$KE = \frac{1}{2}mv^2$$). We classify collisions based on what happens to the kinetic energy.
Elastic Collisions
An elastic collision is a "perfect bounce".
- Momentum is conserved.
- Kinetic energy is also conserved.
No energy is lost to heat, sound, or bending the objects. While no large-scale collision is perfectly elastic, the collisions between billiard balls or atoms are very close.
Inelastic Collisions
This is any collision where kinetic energy is lost.
- Momentum is conserved.
- Kinetic energy is NOT conserved (it decreases).
The "lost" kinetic energy is converted into other forms, like sound (the 'bang' of the crash), heat (things get hot), and work done to deform the objects (dents in a car).
Perfectly Inelastic Collisions
This is a special, easy-to-handle type of inelastic collision.
- The objects stick together after the collision and move with a single, common final velocity.
- This is where the maximum possible kinetic energy is lost.
Examples: A piece of clay hitting a wall and sticking to it; two train cars coupling together.
The conservation of momentum equation becomes much simpler:
Collision Quick Summary
Elastic Collision: Momentum Conserved? Yes. Kinetic Energy Conserved? Yes.
Inelastic Collision: Momentum Conserved? Yes. Kinetic Energy Conserved? No (KE is lost).
Perfectly Inelastic Collision: Momentum Conserved? Yes. Kinetic Energy Conserved? No (Objects stick together).
Key Takeaway: For ALL collisions in a closed system, momentum is your reliable, conserved friend. Kinetic energy is only conserved in perfect "elastic" bounces.
5. How to Solve 1D Momentum Problems
Let's put it all together. Follow these steps and you'll master these problems in no time!
Your Step-by-Step Guide
1. Draw a Diagram: Sketch a simple "before" and "after" picture of the collision.
2. Define Your Direction: Choose a direction to be positive (+). The opposite direction is then negative (-). (e.g., Right is +, Left is -). This is the MOST CRITICAL step!
3. List Your Variables: Write down the values for m₁, u₁, m₂, u₂, etc. Assign the correct + or - sign to all velocities. Identify what you need to find.
4. Choose Your Equation: Will you use the conservation of momentum equation? $$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$. Or if they stick together? $$m_1 u_1 + m_2 u_2 = (m_1 + m_2) v_{final}$$.
5. Substitute and Solve: Carefully plug in your values, keeping the signs. Solve for the unknown variable.
6. Check Your Answer: Does the sign of your final answer make sense? If you get a negative velocity, it just means the object is moving in the negative direction you defined.
Worked Example: The Sticky Collision
A 3 kg cart (Cart A) moving right at 2 m/s collides with a 1 kg cart (Cart B) that is at rest. They stick together after the collision. What is their final velocity?
Step 1 & 2: Diagram and Direction
Let's define Right as positive (+).
Before: Cart A (3kg) ---> (u_A = +2 m/s). Cart B (1kg) is still (u_B = 0 m/s).
After: Carts A+B (3+1=4kg) move together ---> (v_final = ?).
Step 3: List Variables
m_A = 3 kg
u_A = +2 m/s
m_B = 1 kg
u_B = 0 m/s
v_final = ?
Step 4: Choose Equation
They stick together, so it's a perfectly inelastic collision.
$$ m_A u_A + m_B u_B = (m_A + m_B) v_{final} $$
Step 5: Substitute and Solve
$$ (3)(+2) + (1)(0) = (3 + 1) v_{final} $$
$$ 6 + 0 = (4) v_{final} $$
$$ 6 = 4 v_{final} $$
$$ v_{final} = \frac{6}{4} = 1.5 \text{ m/s} $$
Step 6: Check Answer
The final velocity is +1.5 m/s. The positive sign means the combined carts move to the right, which makes perfect sense! Great job!