page_title: Kinematics
section_title: Relative acceleration
hierarchical_section_title: Kinematics / Velocity and speed / Relative acceleration
caption_attribution_description: English: A Velocity-time graph for physics
context_page_description: Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies, and systems of bodies without considering the forces that cause them to move. Kinematics, as a field of study, is often referred to as the "geometry of motion" and is occasionally seen as a branch of mathematics. A kinematics problem begins by describing the geometry of the system and declaring the initial conditions of any known values of position, velocity and/or acceleration of points within the system. Then, using arguments from geometry, the position, velocity and acceleration of any unknown parts of the system can be determined. The study of how forces act on bodies falls within kinetics, not kinematics.
  Kinematics is used in astrophysics to describe the motion of celestial bodies and collections of such bodies. In mechanical engineering, robotics, and biomechanics kinematics is used to describe the motion of systems composed of joined parts such as an engine, a robotic arm or the human skeleton.
context_section_description: The acceleration of one point C  relative to another point B is simply the difference between their accelerations.
  which is the difference between the components of their accelerations.
  If point C has acceleration components 
  and point B has acceleration components 
  then the acceleration of point C relative to point B is the difference between their components: 
  Alternatively, this same result could be obtained by computing the second time derivative of the relative position vector PB/A.
  Assuming that the initial conditions of the position, , and velocity  at time  are known, the first integration yields the velocity of the particle as a function of time.
  A second integration yields its path (trajectory),
  Additional relations between displacement, velocity, acceleration, and time can be derived. Since the acceleration is constant,  
   can be substituted into the above equation to give:
  A relationship between velocity, position and acceleration without explicit time dependence can be had by solving the average acceleration for time and substituting and simplifying
  where ∘ denotes the dot product, which is appropriate as the products are scalars rather than vectors. 
  The dot can be replaced by the cosine of the angle α between the vectors and the vectors by their magnitudes, in which case:
  In the case of acceleration always in the direction of the motion and the direction of motion should be in positive or negative, the angle between the vectors (α) is 0, so , and
  This can be simplified using the notation for the magnitudes of the vectors  where  can be any curvaceous path taken as the constant tangential acceleration is applied along that path, so
  
  This reduces the parametric equations of motion of the particle to a cartesian relationship of speed versus position. This relation is useful when time is unknown. We also know that  or  is the area under a v, t graph . 
   We can take  by adding the top area and the bottom area. The bottom area is a rectangle, and the area of a rectangle is the  where  is the width and  is the height. In this case  and  (note that the  here is different from the acceleration ). This means that the bottom area is . Now let's find the top area (a triangle). The area of a trangle is  where  is the base and  is the height. In this case,  &  or . Adding  and  results in the equation  results in the equation . This equation is very useful when the final velocity v is unknown.
caption: Velocity Time physics graph
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