How fast can things go?

In this area of study students use Newton’s laws of motion to analyse relative motion, circular motion and projectile motion. Newton’s laws of motion give important insights into a range of motion both on Earth and beyond. At very high speeds, however, these laws are insufficient to model motion and Einstein’s theory of special relativity provides a better model. Students compare Newton’s and Einstein’s explanations of motion and evaluate the circumstances in which they can be applied. They explore the relationships between force, energy and mass.

On completion of this unit the student should be able to investigate motion and related energy transformations experimentally, analyse motion using Newton’s laws of motion in one and two dimensions, and explain the motion of objects moving at very large speeds using Einstein’s theory of special relativity.

Key knowledge

Newton’s laws of motion

  • investigate and apply theoretically and practically Newton’s three laws of motion in situations where two or more coplanar forces act along a straight line and in two dimensions
  • investigate and analyse theoretically and practically the uniform circular motion of an object moving in a horizontal plane: \left (F_{net}=\frac{mv^2}{r}  \right ) , including:
    – a vehicle moving around a circular road
    – a vehicle moving around a banked track
    – an object on the end of a string
  • model natural and artificial satellite motion as uniform circular motion
  • investigate and apply theoretically Newton’s second law to circular motion in a vertical plane (forces at the highest and lowest positions only)
  • investigate and analyse theoretically and practically the motion of projectiles near Earth’s surface, including a qualitative description of the effects of air resistance
  • investigate and apply theoretically and practically the laws of energy and momentum conservation in isolated systems in one dimension.

Einstein’s theory of special relativity

  • describe Einstein’s two postulates for his theory of special relativity that:
    – the laws of physics are the same in all inertial (non-accelerated) frames of reference
    – the speed of light has a constant value for all observers regardless of their motion or the motion of the source
  • compare Einstein’s theory of special relativity with the principles of classical physics
  • describe proper time (t_0 ) as the time interval between two events in a reference frame where the two events occur at the same point in space
  • describe proper length (L_0 ) as the length that is measured in the frame of reference in which objects are at rest
  • model mathematically time dilation and length contraction at speeds approaching c using the equations:t=t_0\gamma and L=\frac{L_0}{\gamma} where \gamma=\left ( {1-\frac{v^2}{c^2}} \right )^{-{\frac{1}{2}}
  • explain why muons can reach Earth even though their half-lives would suggest that they should decay in the outer atmosphere.

Relationships between force, energy and mass

  • investigate and analyse theoretically and practically impulse in an isolated system for collisions between objects moving in a straight line: F\Delta t=m\Delta v
  • investigate and apply theoretically and practically the concept of work done by a constant force using:
    – work done = constant force × distance moved in direction of net force
    – work done = area under force-distance graph
  • analyse transformations of energy between kinetic energy, strain potential energy, gravitational potential energy and energy dissipated to the environment (considered as a combination of heat, sound and deformation of material):
    – kinetic energy at low speeds: E_k=\frac{1}{2}mv^2 ; elastic and inelastic collisions with reference to conservation of kinetic energy
    – strain potential energy: area under force-distance graph including ideal springs obeying Hooke’s Law: E_s=\frac{1}{2}k\Delta x^2
    – gravitational potential energy: E_g=mg\Delta h or from area under a force-distance graph and area under a field-distance graph multiplied by mass
  • interpret Einstein’s prediction by showing that the total ‘mass-energy’ of an object is given by: E_{tot}=E_k+E_0={\gamma}mc^2  where E_{0}=mc^2 , and where kinetic energy can be calculated by: E_{k}=(\gamma-1)mc^2
  • describe how matter is converted to energy by nuclear fusion in the Sun, which leads to its mass decreasing and the emission of electromagnetic radiation.


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