A topic in Learners' conceptions and thinking
Newton's second law is a common source of learning difficulties and alternative conceptions.
(Read about common alternative conceptions.)
Newton-2: The law
Newton's first law (N2) is concerned with how a force changes a body's motion.
- The rate of change of momentum is proportional to the magnitude of the applied force, and takes place in the direction of the applied force.
As an object's mass is usually constant* the rate of change of momentum is equal to the acceleration.
(*Strictly, a body's mass increases with its speed (according to the special theory of relativity), but this effect is negligible unless the motion is approaching light speeds.)
dmv/dt = mdv/dt [if mass is constant] = ma
so, F=ma
This then leads to a seemingly simple equation. However,
This simplification only work is there is a steady force – if the force is changing, dv/dt will be changing and there will be no constant value of a. In introductory physics examples are usually limited to steady forces, but many relativist examples are not like this (e.g. when a tennis ball hits a tennis ball the force increase then decreases).
It is also important to note that F and a are vectors, so ' F=ma' reflects how that the acceleration will occur in the direction of the force – but learners may not appreciate this unless the point is regularly reinforced.
The counter-intuitive nature of the second law
This leads to one of the common learning difficulties which is that learners may often assume that after a force is applied, then the motion will be in the direction of the force. This will only be so in special cases:
- if the object is initially stationary, then it will accelerate in the direction of the applied force and so move in the direction of the force.
- if the force is applied in the direction in which the body is already moving, then it will accelerate in that direction – and just go faster
Another special case is when the force is applied against the direction of motion. This is quite common as resistive forces (friction, air resistance) act against motion.
if the force is applied against the direction in which the body is already moving, then it will be retarded and decelerate
(If the applied force is sufficiently large the object may decelerate and then move off in the opposite direction. To a physicists these last three cases can all be seen as an acceleration in the direction of motion: but in the second case a negative acceleration so the change in velocity is negative; in the third case a sufficient negative acceleration such that the change in velocity is so negative that it has greater magnitude than the original velocity.)
In the general case (when a force acts on the body, but not along the trajectory of it initial motion) the final direction is different to both the initial velocity and the direction of the applied force. For many learners this is not obvious!
This may become obvious if we set up a convincing thought experiment.
Imagine a blue whale swimming in a straight line at a constant depth of 10m beneath the ocean surface. (i.e., it has constant velocity.)
Imagine that something hits the side of the whale applying a force perpendicular to its direction of travel.
Will the direction of the whale change? (Yes, because a force has been applied.)
How much will the whale be pushed off course if it had been hit by a submarine?
How much will the whale be pushed off course if it had been hit by a shark?
How much will the whale be pushed off course if it had been hit by a sardine?
No sea organisms were hurt in this thought experiment
Producing curved paths
An object with a constant velocity (and so moving along a straight path) that is subject to a continuing force will continue to change momentum, and so continue to change direction as long as the direction of the force is not aligned with its direction.
Three particular cases.
Projectile motion.
An object moving through the air and subject to the constant effect of the earth's gravitational field will follow a parabolic path. The downward force due to its weight will cause it to accelerate downward, but will not effect its forward motion.
Falling object
At least in the ideal case. As air resistance will retard its forward motion it will eventually have no forward and only downwards motion. However the transition will be gradual – unlike in cartoons where a body suddenly changes direction to fall downwards (usually at the point where the character realises they have run over the edge of a cliff!)
Circular motion
If the applied force changes direction as the object changes direction such that the force is always acting perpendicular to the motion then the body will move in a circular arc.
This may sound unlikely to some learners, but:
- in particle accelerators, powerful magnets may be arrange to bend a beam of charged particles to make them travel around a circuit
- moons orbit planets, and planets orbit stars in elliptical paths (some very nearly circular). The gravitational force is always perpendicular to a circular orbit as a radius (the force direction) is always perpendicular to the circumference of a circle (the orbit)
Learning difficulties
When an object is dejected form a moving body learners often ignore that it initially has momentum in the original direction. So, an object thrown out of a car window is (antisocial and potentially dangerous and) often considered to fall down as if dropped form a stationary position. Parachutists are assumed to fall vertically form a plane.
One way to help learners we-organise their experience here is to play at (sorry, simulate) planes. Set up a target on the floor, and have students approach at moderate velocity and release their load of famine relief supplies to land on the target. If they wait till they are directly over the target they will overshoot!
Circular motion is a special challenge for many learners who may think it is a 'natural motion' that does not need to be explained buy a force. Those who do recognise a force may assume it acts around the orbit (in the direction of travel) rather than towards the centre.
Many learners think that circular motion is a balanced (for example that a centripetal force {the actual force acting} is balanced by a centrifugal force {an imaginary force}). This may draw on a misunderstanding of Newton's third law as suggesting that every force is balanced by another.
Read about conceptions of Newton's third law
If an object is projected vertically up into the the air then it will experience a constant downward force that will first decrease its upwards velocity and then give it a downward velocity. Learners often see this as a multi-stage process with the instant at the top of the path as having some special significance.
As students often associate the force acting with the direction of motion this may be understood as three stage process: the body starts with net upward force, which is gradually used up, till at the top there is no overall force acting (if this were so it would float there!) and then gravity acting on the way down.
Alternative conceptions relating to Newton-2: examples
The book Student Thinking and Learning in Science: Perspectives on the Nature and Development of Learners' Ideas gives an account of the nature of learners' conceptions, and how they develop, and how teachers can plan teaching accordingly.
It includes many examples of student alternative conceptions in science topics.
