Rectilinear Kinematics: Continuous 12Motion

Rectilinear Kinematics: Continuous 12Motion

1. Rectilinear Kinematics

   • The kinematics of a particle is characterized by specifying, at any given instant, the particle’s position, velocity, and acceleration.

2. Position

   • The straight-line path of a particle will be defined using a single coordinate axis s, Fig. 12–1a. The origin O on the path is a fixed point, and from this point the position coordinate s is used to specify the location of the particle at any given instant. The magnitude of s is the distance from O to the particle, usually measured in meters (m) or feet (ft), and the sense of direction is defined by the algebraic sign on s.

3. Displacement

   • The displacement of the particle is defined as the change in its position. For example, if the particle moves from one point to another, Fig. 12–1b, the displacement is s = s - s

4. Velocity

   • If the particle moves through a displacement s during the time interval t, the average velocity of the particle during this time interval is vavg = s t If we take smaller and smaller values of t, the magnitude of s becomes smaller and smaller. Consequently, the instantaneous velocity is a vector defined as v = lim tS0 (s>t), or (S+ ) v = ds dt

5. Acceleration

   • Provided the velocity of the particle is known at two points, the average acceleration of the particle during the time interval t is defined as aavg = v t Here v represents the difference in the velocity during the time interval t, i.e., v = v - v, Fig. 12–1e. The instantaneous acceleration at time t is a vector that is found by taking smaller and smaller values of t and corresponding smaller and smaller values of v, so that a = lim tS0 (v>t), or (S+ ) a = dv dt

6. Constant Acceleration, a = ac .

7. Velocity as a Function of Time. Integrate ac = dv>dt, assuming that initially v = v0 when t = 0. L v v0 dv = L t 0 ac dt

8. Position as a Function of Time. Integrate v = ds>dt = v0 + act, assuming that initially s = s0 when t = 0. L s s0 ds = L t 0 (v0 + act) dt

9. Velocity as a Function of Position. Either solve for t in Eq. 12–4 and substitute into Eq. 12–5, or integrate v dv = ac ds, assuming that initially v = v0 at s = s0 . L v v0 v dv = L s s0 ac ds