A model of accelerated motion

The last step in building a model for motion is to develop a single equation that relates position, velocity, time, and acceleration. Consider a moving object with initial velocity v0 that undergoes constant acceleration. At time t, the velocity has increased from v0 to v. The distance the object travels between time t = 0 and time t is the area shaded on the graph. Read the text aloud
Total distance traveled is the area under the curve—in this case the sum of the area of the triangle and rectangle
The v vs. t graph breaks down into two shapes. The area of a triangle is ½ base × height. For triangle A this is ½(vv0)t. We also know that the change in velocity is acceleration × time, so v – v0 = at. Therefore, the area of triangle A is ½at2. The area of rectangle B is v0t. Area on a v vs. t graph equals distance, and adding the triangle to the rectangle gives us this result
d= v 0 t+ 1 2 a t 2
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One last step remains. The distance traveled is d = x – x0. Substituting this expression yields equation (4.3), which relates position x at any time t to initial position x0, initial velocity v0, and acceleration a. Read the text aloud
(4.3) x= x 0 + v 0 t+ 1 2 a t 2
x  = position (m)
x0  = initial position (m)
v0  = initial speed (m/s)
a  = acceleration (m/s2)
t  = time (s)
accelerated motion
Equation (4.3) has three terms on the right-hand side, and each term has its own meaning. The first term is the initial position. The second term is the change in position the object would have had if it continued at constant initial speed v0. The third term is the additional change in position resulting from changes in speed that come from acceleration. Note that, if the acceleration is zero, we get back x = x0 + vt, the equation for constant velocity from the last chapter! Read the text aloud
Understanding the different terms in the equation for position as a function of time
Show Interpreting acceleration versus time graphs
A baseball is thrown straight upward and returns to the point from which it was thrown after 5.0 s. Find the baseball’s original speed. The acceleration of the baseball is 9.8 m/s2 downward.
  1. 49 m/s
  2. 4.9 m/s
  3. 123 m/s
  4. 25 m/s

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