002 rectiliear kinematics continuous motion
- 1. RECTILINEAR KINEMATICS: Continuous Motion The kinematics of a particle is characterized by specifying, at any given instant, the particle’s position, velocity, and acceleration. INSTANTANEOUS SPEED Consider a car starting from rest and note the time at which the odometer shows it has covered 100 m, 200 m, and so on. The data is shown in the table below. Find the instantaneous speed of the car at t = 40s. Table 1: Total Distance, m 0 100 200 300 400 500 Time Elapsed, s 0 28 40 49 57 63 CONSTANT SPEED When the instantaneous speed of an object does not change, it is moving at constant speed. The distance-time graph of a car travelling at a constant speed is a straight line, a typical graph is shown in the figure below.
- 2. SPEED vs. VELOCITY The speed of a moving object specified only how fast the object is going, regardless of its direction. A complete description that includes both speed and direction is called velocity. Because the rate of change of distance tells us only “how fast,” the definition of velocity involves a different quantity, displacement. The displacement of an object that has moved is its change in position. ACCELERATION In the real world few objects move at constant velocity for very long. An object whose velocity is increasing or decreasing, or whose direction is changed, is said to be accelerated. Table 2 shows the instantaneous speeds of the car at 10-s intervals, given in Figure 2. Find the acceleration at t=20-s and t=50-s. Table 2: Time Elapsed, s 0 10 20 30 40 50 60 Instantaneous speed, m/s 0 2.5 5 7.5 10 12.5 15
- 3. Because the value of the acceleration a is the same, the acceleration is constant. If the acceleration varied, we would get different values of a at different times. Example 1: The displacement, s of a particle that is confined to move along a straight line is given by s = 2t3 – 24t + 6, where s is measured in meters from a convenient origin and t is in seconds. Determine (a) The time required for the particle to reach a velocity of 72 m/s from its initial condition at t=0. (b) The acceleration of the particle when v=30m/s. (c) The net displacement of the particle during the interval from t=1s to t=4s. (d) Plot the displacement, velocity and acceleration diagrams. Solution: