3 Motion in One Dimension The study of motion without regard to the forces that influence the motion is called kinematics. If an object is at position x1 at time t1 and at x2 at time t2, the displacement vector ∆x = x2 −x1. The average velocity v is defined as v = x2−x1 t2 −t1 = ∆x ∆t. If an object experiences a displacement x in a time t, its instantaneous velocity is: vi = lim ∆t→0 ∆x ∆t = dx dt . Instantaneous velocity is the limit of the average velocity as the time interval approaches zero; it equals the instantaneous rate of change of position with time. Velocity is a vector and its magnitude is speed. Speed and velocity are measured in metres per second (m/s or ms−1). Velocity is the slope of a graph of displacement x versus time t. Note: The terms ’speed’ and ’velocity’ are used interchangeably in everyday language, but they have distinct definitions in Physics. We use the terms speed to denote distance travelled divided by time, on either an average or an instantaneous basis. Instantaneous speed measures how fast a particle is moving; instantaneous velocity measures how fast and in what direction it’s moving. For example, a particle with instantaneous velocity v = 25m/s and a second particle with v = −25m/s are moving in opposite directions at the same instantaneous speed 25 m/s. Instantaneous speed is the magnitude of instantaneous velocity, and so instantaneous speed can never be negative. Note however that average speed is not the magnitude of average velocity. Acceleration: When the velocity of a moving body changes with time, the body is said to have acceleration. Acceleration describes the rate of change of velocity with time, and it is a vector quantity. Average acceleration is defined as: a = v2−v1 t2 −t1 = ∆v and instantaneous acceleration is defined as: ax = lim ∆t→0 ∆t ∆v ∆t = dv dt . The instantaneous acceleration is the limit of the average acceleration as the time interval approaches zero. The SI unit of acceleration is metre per second squared (m/s2). Acceleration is the second derivative of displacement. The slope of a graph of velocity against time gives the average acceleration. Also, on a graph of velocity as a function of time, the instantaneous acceleration at any point is equal to the slope of the tangent to the curve at that point. If a car accelerates from 15m/s to 35m/s in 5sec, then average acceleration a = 20−30 5=−2m/s2. For motion on a straight line, the acceleration os in the direction of that line. 3.1 Motion with constant acceleration When the acceleration is constant, it is easy to derive equations for position x and velocity v as functions of time. If we replace the average acceleration (from its definition) by the constant (instantaneous) acceleration, a, we have: a = v2−v1 t2 −t1
