Note Relationship between s, v and at The quantities s, v and at are related in the same manner as the quantities s, u and a for rectilinear motion. In particular we have that v=s, at= i, and at ds= v du. This means that if we have a way of knowing at ve may be able to integrate the tangential component of the motion independently. We will be exploiting these relations in the future The vectors et and en, and their respective coordinates t and n, define two orthogonal directions. The plane defined by these two directions, is called the osculating plane. This plane changes from point to point, and can be thought of as the plane that locally contains the trajectory(Note that the tangent is the current direction of the velocity, and the normal is the direction into which the velocity is changing In order to define a right handed set of axes we need to introduce an additional unit vector which is orthogonal to et and en. This vector is called the binormal, and is defined as eb=et x en At any point in the trajectory, the position vector, the velocity and acceleration can be referred to these axes. In particular, the velocity and acceleration take very simple forms Uet+—e The difficulty of working with this reference frame stems from the fact that the orientation of the axis depends on the trajectory itself. The position vector, r, needs to be found by integrating the relation dr/dt= v as follows dt where ro =r(o) is given by the initial condition We note that, by construction, the component of the acceleration along the binormal is always zero When the trajectory is planar, the binormal stays constant(orthogonal to the plane of motion).However hen the trajectory is a space curve, the binormal changes with s. It can be shown(see note below)that the derivative of the binormal is always along the direction of the normal. The rate of change of the binormalNote Relationship between s, v and at The quantities s, v and at are related in the same manner as the quantities s, v and a for rectilinear motion. In particular we have that v = ˙s, at = ˙v, and at ds = v dv. This means that if we have a way of knowing at, we may be able to integrate the tangential component of the motion independently. We will be exploiting these relations in the future. The vectors et and en, and their respective coordinates t and n, define two orthogonal directions. The plane defined by these two directions, is called the osculating plane. This plane changes from point to point, and can be thought of as the plane that locally contains the trajectory (Note that the tangent is the current direction of the velocity, and the normal is the direction into which the velocity is changing). In order to define a right handed set of axes we need to introduce an additional unit vector which is orthogonal to et and en. This vector is called the binormal, and is defined as eb = et × en. At any point in the trajectory, the position vector, the velocity and acceleration can be referred to these axes. In particular, the velocity and acceleration take very simple forms, v = vet a = ˙vet + v 2 ρ en . The difficulty of working with this reference frame stems from the fact that the orientation of the axis depends on the trajectory itself. The position vector, r, needs to be found by integrating the relation dr/dt = v as follows, r = r0 + Z t 0 v dt , where r0 = r(0) is given by the initial condition. We note that, by construction, the component of the acceleration along the binormal is always zero. When the trajectory is planar, the binormal stays constant (orthogonal to the plane of motion). However, when the trajectory is a space curve, the binormal changes with s. It can be shown (see note below) that the derivative of the binormal is always along the direction of the normal. The rate of change of the binormal 5