Chapter 2 One-Dimensional Kinematics

2.10 New – anti-derivatives and motion

$$\frac {linear}{206,256}=\frac{linear}{distance} [/frac]  $$

Learning Objectives

  • Derive the kinematic equations for constant acceleration using integral calculus.
  • Use the integral formulation of the kinematic equations in analyzing motion.
  • Find the functional form of velocity versus time given the acceleration function.
  • Find the functional form of position versus time given the velocity function.

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This section assumes you have enough background in calculus to be familiar with integration. In Instantaneous Velocity and Speed and Average and Instantaneous Acceleration we introduced the kinematic functions of velocity and acceleration using the derivative. By taking the derivative of the position function we found the velocity function, and likewise by taking the derivative of the velocity function we found the acceleration function. Using integral calculus, we can work backward and calculate the velocity function from the acceleration function, and the position function from the velocity function.

 

Kinematic Equations from Integral Calculus

Let’s begin with a particle with an acceleration a(t) is a known function of time. Since the time derivative of the velocity function is acceleration,

 

x(6.3)=5.0(6.3)−124(6.3)3=21.1m.(3.6.15)(3.6.15)x(6.3)=5.0(6.3)−124(6.3)3=21.1m.
Graph A is a plot of velocity in meters per second as a function of time in seconds. Velocity is five meters per second at the beginning and decreases to zero. Graph B is a plot of position in meters as a function of time in seconds. Position is zero at the beginning, increases reaching maximum between six and seven seconds, and then starts to decrease.
Figure 3.6.13.6.1: (a) Velocity of the motorboat as a function of time. The motorboat decreases its velocity to zero in 6.3 s. At times greater than this, velocity becomes negative—meaning, the boat is reversing direction. (b) Position of the motorboat as a function of time. At t = 6.3 s, the velocity is zero and the boat has stopped. At times greater than this, the velocity becomes negative—meaning, if the boat continues to move with the same acceleration, it reverses direction and heads back toward where it originated.

 

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