Mechanics
Newton's Laws of Motion and Momentum
5th Year · 6th Year (Leaving Cert)
- ✓By the end of this lesson students will be able to state and apply Newton's three laws of motion.
- ✓By the end of this lesson students will be able to define momentum and calculate it for a moving object.
- ✓By the end of this lesson students will be able to state and apply the Principle of Conservation of Momentum.
- ✓By the end of this lesson students will be able to define impulse and relate it to the change in momentum.
- ✓By the end of this lesson students will be able to solve problems involving Newton's laws, momentum, and impulse.
Key concepts
An object will remain at rest or in uniform motion in a straight line unless acted upon by an external resultant force. This law introduces the concept of inertia, which is the resistance of an object to a change in its state of motion. A resultant force is the single force that has the same effect as all the individual forces acting on an object.
The rate of change of an object's momentum is directly proportional to the resultant force applied to it and acts in the direction of the force. Mathematically, this is often expressed as F = ma, where F is the resultant force (in Newtons, N), m is the mass of the object (in kilograms, kg), and a is its acceleration (in metres per second squared, m s⁻²). This formula is derived from F = Δp/Δt, assuming constant mass. Force and acceleration are vector quantities.
If object A exerts a force on object B, then object B exerts an equal and opposite force on object A. These forces are known as action-reaction pairs. It is crucial to remember that these forces always act on different objects and therefore do not cancel each other out.
Momentum (p) is a measure of the 'quantity of motion' an object possesses. It is defined as the product of an object's mass (m) and its velocity (v). Momentum is a vector quantity, meaning it has both magnitude and direction. Its direction is the same as the object's velocity. The SI unit for momentum is kilogram metre per second (kg m s⁻¹).
In any collision or explosion involving two or more objects, the total momentum of the system before the event is equal to the total momentum of the system after the event, provided no external resultant forces act on the system. This principle applies to isolated systems where there is no net external force. For two objects, it can be stated as: total momentum before = total momentum after.
Impulse is defined as the product of the force (F) acting on an object and the time (t) for which the force acts. It is also equal to the change in momentum (Δp) of the object. Impulse is a vector quantity and its direction is the same as the force. The SI unit for impulse is Newton second (N s), which is equivalent to kilogram metre per second (kg m s⁻¹). Impulse is important in understanding safety features like airbags, which increase the time of impact to reduce the force.
Key facts to remember
- 1Newton's First Law describes inertia: an object resists changes to its state of motion.
- 2Newton's Second Law, F = ma, quantifies the relationship between force, mass, and acceleration.
- 3Newton's Third Law states that action and reaction forces are equal and opposite, acting on different objects.
- 4Momentum (p = mv) is a vector quantity, with units of kg m s⁻¹.
- 5The Principle of Conservation of Momentum states that total momentum is conserved in an isolated system.
- 6Impulse (Ft) is equal to the change in momentum (Δp = mv - mu) and has units of N s or kg m s⁻¹.
- 7All forces, velocities, and momentum values must be treated as vectors, considering their direction.
Worked examples
Example 1
A car of mass 1200 kg accelerates uniformly from rest to 20 m s⁻¹ in 8.0 seconds. Calculate the resultant force acting on the car.
Answer
F = 3000 N
Remember to use SI units throughout your calculations.
Example 2
A trolley of mass 2.0 kg moving at 3.0 m s⁻¹ collides head-on with a stationary trolley of mass 4.0 kg. After the collision, the two trolleys stick together. Calculate their common velocity after the collision.
Answer
v = 1.0 m s⁻¹
In collision problems, always define a positive direction and be consistent with signs for velocities.
Example 3
A hurley strikes a sliotar (mass 0.12 kg) initially at rest, giving it a velocity of 30 m s⁻¹. If the contact time between the hurley and sliotar is 0.005 seconds, calculate the average force exerted on the sliotar.
Answer
F = 720 N
The impulse is a vector, so the force will be in the same direction as the change in momentum.
Common mistakes
- ✗Confusing action-reaction forces (Newton's 3rd Law) as acting on the same object, leading to incorrect cancellation.
- ✗Forgetting that momentum and impulse are vector quantities and not accounting for direction (e.g., using positive and negative signs for velocities).
- ✗Not converting units to SI units (e.g., grams to kilograms, km/h to m/s) before performing calculations.
- ✗Applying the Principle of Conservation of Momentum when there are significant external resultant forces acting on the system.
- ✗Mixing up the definitions or units of force, momentum, and impulse.
Exam tips
- ★Always draw a clear diagram for collision problems, indicating the direction of velocities before and after the event.
- ★Choose a consistent positive direction for all vector quantities (velocity, momentum, force) at the start of your calculation.
- ★List all known and unknown quantities with their correct units before attempting to solve a problem.
- ★Show all steps of your working clearly, including the formula used and any unit conversions, as marks are awarded for method.
- ★Check the units in your final answer to ensure they are appropriate for the quantity being calculated.
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