Speed and acceleration – linear motion.
Level 0 (green)- this is basic material that you have probably encountered already, although the approach may be slightly different. No prior knowledge is assumed. This section deals simply with motion along a straight line.
The velocity topic is a more sophisticated version of this topic that uses vectors.
Before analysing movement we need to have a quantitative way of specifying the location of a particle or of a system of particles.
For motion in one dimension, the simplest way to this is to define the position of a particle relative to some origin as a Cartesian coordinate x.
The speed of a particle is the rate of change of the position,
mathematically, it is the derivative of the position with respect to time.
This quantity is a vector, which for linear motion simply means that the sign has physical significance. If it is positive the particle is moving in the direction of increasing x, and if it is negative in the opposite direction.
Momentum is an important quantity related to velocity, the momentum of a particle is defined as
p = mv.
The symbol p is the internationally recommended symbol in Physics to denote momentum.
Because the velocity is a vector, so is the momentum, and the sign has the same interpretation. The momentum reflects the everyday notion that it is easier to stop or deflect a light particle (e.g. a ping-pong ball) travelling with a given velocity than a heavy one (e.g. a train).
Momentum is closely associated with the concept of force and Newton's laws.
Since the velocity is the derivative of the position, the position of a particle may be obtained by integrating the velocity.
The right hand side of this equation is the displacement of the particle from its position at time t1.
As a simple example, for a body travelling with a constant speed v, the displacement in a time period t would be vt.
The acceleration is the time derivative of the velocity.
The velocity may therefore be regained by integrating the acceleration:
Example: constant acceleration formulas
A common example in A level Physics is the formulas that apply to the special case of motion in one dimension with constant acceleration, for example a body falling under the influence of gravity, or an ion being accelerated by a constant electric field.
Because we are in one dimension we do not need to worry about vectors, and treat the acceleration a as a constant.
We can find the velocity of the particle at any time by integrating the acceleration:
and we can find the displacement of the particle from its initial position by integrating the velocity:
The third useful equation for constant acceleration,
may be derived by suitable substitutions from the other two equations, but is much more naturally approached using the conservation of energy.