2.5 Momentum
Momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. Every moving object has momentum, which depends directly on both its mass and its velocity. An object with a larger mass or moving at a higher velocity is more difficult to stop because it possesses a greater momentum. Furthermore, when objects interact through collisions or explosions in a closed system with no external forces, their total momentum remains conserved.
Learning Objectives
- Define momentum as the product of mass and velocity (\(p = mv\)) and identify it as a vector quantity.
- State the S.I. unit for momentum as \(\text{kg m s}^{-1}\) or \(\text{N s}\).
- State the Principle of Conservation of Momentum: total momentum before collision equals total momentum after collision if no external force acts on the system.
- Apply the Principle of Conservation of Momentum to situations involving collisions (\(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\)) and explosions (\(0 = m_1v_1 + m_2v_2\) or \(m_1v_1 = -m_2v_2\)).
- Describe daily life applications and situations related to momentum and the conservation of momentum, such as fire hoses and launching rockets.
Principle of Conservation of Momentum in Collisions and Explosions
Compare how momentum is conserved when objects collide and when a stationary object explodes into parts moving in opposite directions.
Collision
Two spheres approach each other with velocities \(u_1\) and \(u_2\), then separate with velocities \(v_1\) and \(v_2\).
\(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\)
Explosion
A rocket at rest (\(p = 0\)) splits into downward exhaust gases (\(m_1v_1\)) and an upward rocket body (\(m_2v_2\)).
\(m_1v_1 = -m_2v_2\)
Short Explanation
What is Momentum?
Momentum is the product of an object's mass and its velocity. It is a vector quantity, so its direction is the same as the direction of the object's velocity.
Formula and S.I. Unit
\(p = mv\) where \(p\) is momentum, \(m\) is mass and \(v\) is velocity. The S.I. unit is \(\text{kg m s}^{-1}\) or \(\text{N s}\).
Principle of Conservation of Momentum
The total momentum before collision is equal to the total momentum after collision if no external force is acting on the system.
Explosion
An explosion refers to a situation where an object at rest breaks up into two or more parts. Before explosion, total momentum is \(0\) because the object is at rest.
Collision formula
\(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\)If two objects stick together after collision, they share a common velocity \(v\): \(m_1u_1 + m_2u_2 = (m_1 + m_2)v\).
Explosion formula
\(0 = m_1v_1 + m_2v_2\)This can be written as \(m_1v_1 = -m_2v_2\). The negative sign indicates that the two parts move in opposite directions.
| Quantity | Symbol | S.I. Unit |
|---|---|---|
| Momentum | \(p\) | \(\text{kg m s}^{-1}\) or \(\text{N s}\) |
| Mass | \(m\) | \(\text{kg}\) |
| Velocity | \(v\) | \(\text{m s}^{-1}\) |
Worked Example: Calculating Momentum
A lorry of mass \(20\,000\text{ kg}\) moves at \(22\text{ m s}^{-1}\). Its momentum is \(p = mv = 20\,000\text{ kg} \times 22\text{ m s}^{-1} = 440\,000\text{ kg m s}^{-1} = 440\,000\text{ N s}\).
Daily Life Applications
Fire hose: Water gushing out of the hose at high speed has a high forward momentum. To balance this backward reaction momentum, two or more firemen are needed to hold the hose tightly.
Try to Answer First
Answer in your mind, then press “Check Answer”.
What is the momentum of an \(800\text{ kg}\) car moving with a velocity of \(25\text{ m s}^{-1}\)?
Check Answer
State the condition required for the total momentum of a system to be conserved.
Check Answer
A lorry of mass \(1\,000\text{ kg}\) moving at \(5.0\text{ m s}^{-1}\) collides with a car of mass \(800\text{ kg}\) moving at \(2.0\text{ m s}^{-1}\) in the same direction. If the lorry continues moving at \(3.4\text{ m s}^{-1}\) in the same direction after the collision, calculate the velocity of the car.
Check Answer
\((1000)(5.0) + (800)(2.0) = (1000)(3.4) + (800)(v_2)\)
\(5000 + 1600 = 3400 + 800v_2\)
\(6600 - 3400 = 800v_2\)
\(3200 = 800v_2 \implies v_2 = 4.0\text{ m s}^{-1}\) in the same direction.
Common Mistakes
- Forgetting that momentum is a vector quantity and omitting direction or sign conventions when velocities are in opposite directions.
- Confusing the units of momentum (\(\text{kg m s}^{-1}\) or \(\text{N s}\)) with force (\(\text{N}\) or \(\text{kg m s}^{-2}\)).
- Assuming momentum is conserved even when significant net external forces (such as friction on an uncompensated track) are present.
- Misinterpreting the negative sign in explosion calculations (\(m_1v_1 = -m_2v_2\)) as a negative mass or magnitude, rather than an opposite direction.
Concept Misunderstandings
A stationary object with an enormous mass has high momentum.
An object at rest has zero velocity (\(v = 0\)), so its momentum is strictly zero, regardless of how massive it is.
In an explosion, momentum is created from nothing because objects suddenly gain speed.
The initial momentum is zero. After the explosion, parts move in opposite directions; when added as vectors, the opposite momenta cancel each other out, keeping the total momentum zero.
The displacement of an obstacle hit by an object depends only on the object's speed.
The displacement depends on both the mass and velocity (the momentum) of the moving object.
Summary
- Momentum (\(p\)): Product of mass (\(m\)) and velocity (\(v\)), \(p = mv\).
- Vector Quantity: Direction of momentum is identical to the direction of velocity.
- S.I. Units: \(\text{kg m s}^{-1}\) or \(\text{N s}\).
- Principle of Conservation of Momentum: Total momentum before an interaction equals total momentum after the interaction, provided no external forces act.
- Collision Formula: \(m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2\)
- Explosion Formula: \(0 = m_1v_1 + m_2v_2 \implies m_1v_1 = -m_2v_2\)
Short Activity
Objective: Identify momentum, conservation of momentum, collisions and explosions.
A. Objective Quiz
1 Which of the following is an S.I. unit for momentum?
2 Under what condition is the total momentum of a colliding system conserved?
B. Fill in the Blanks
3 Isaac Newton described momentum as the "quantity of ".
4 In an explosion where an object initially at rest breaks into two parts of masses \(m_1\) and \(m_2\), the equation relating their momenta after explosion is \(m_1v_1 = \) .
C. Matching / Drag and Drop
Match each momentum term or phenomenon to its correct textbook description. If using a phone, tap the answer first, then tap the matching box.
