2.1 
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Interactive Simulations   Position and displacement in one dimension    Displacement and position in two dimensions    Solve v=dx/dt for dt    Velocity, position, and time (basic version)    Velocity, position, and time (more advanced version)    Acceleration (basic version)    Acceleration (more advanced version)    Static equilibrium    Static equilibrium and the lever    Newton's second law (basic version)    Newton's second law (more advanced version)    Circular motion interactive simulation    Orbits of the planets and dwarf planets    Orbits simulation Interactive Calculators   Speed calculator    Velocity calculator    Acceleration calculator    Velocity in accelerated motion    Average velocity calculator    Hooke's law calculator    Torque calculator    Rotational equilibrium and torque    Newton's second law calculator    Momentum calculator    Angular velocity    Linear velocity and angular velocity    Centripetal acceleration    Centripetal force    Law of universal gravitation calculator
1 - Science of Physics
2 - Forces and Motion      Chapter 2 at a glance      2.1 - Motion      2.2 - Force, momentum, and Newton's laws      2.3 - Circular motion      2.4 - Solving harder physics problems      2.5 - Chapter review3 - Energy Transformations4 - Waves and Sound
5 - Electricity and Magnetism
6 - Light and Optics
7 - The Atom
8 - Linear Motion
9 - Vectors and Forces
10 - Forces
11 - Newton's Laws and Gravitation
12 - Circular Motion and Gravitation
13 - Torque and Static Equilibrium
14 - Momentum and Collisions
15 - Angular momentum
16 - Power and Machines
17 - Harmonic Motion
18 - Waves
19 - Sound
20 - Electricity
21 - Electric and Magnetic Fields
22 - Electromagnetism and Induction
23 - Light and Color
24 - Geometrical Optics
25 - Electromagnetic Radiation
26 - The Properties of Matter
27 - Thermodynamics and Heat Transfer
28 - The Atom
29 - The Quantum Theory
30 - TBD
31 - TBD
32 - TBD
33 - TBD
34 - TBD
35 - TBD
36 - TBD
37 - The atomic nucleus and radioactivity
38 - Nuclear reactions
39 - Relativity
40 - Particle physics and the standard model
41 - Electronics (semiconductors)
42 - Nuclear technology
43 - Lasers and photonics
44 - Metals, alloys and materials science
45 - Communications technology
46 - Buildings and structures
47 - Energy technology
48 - Biophysics
49 - Matter and Energy, Space and Time
50 - Energy Transformations
51 - Nanotechnology
52 - Website Videos
53 - ExtraStuff
55 - Modeling Motion, Friction, and Center of Mass
56 - Work and the Conservation of Energy
How position changes in accelerated motion
How does position change with acceleration? The last step in building a model for motion is to develop a single equation that relates position, velocity, time, and acceleration. Because velocity may not be constant, we use the fact that distance is equal to area on the v vs. t graph. 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.
Total distance traveled is the area under the curve—in this case the sum of the area of the triangle and rectangle
How do we calculate the distance? In the graph are two geometrical shapes: a triangle marked “A” and a rectangle marked “B.” The area of triangle is ½ base × height. On the graph this area is ½(v – v0)t. We also know that the change in velocity is acceleration × time, so v – v0 = at. Substituting this expression shows that the area of the triangle is ½at2. The area of the rectangle is v0t. Adding these two areas together gives us an equation for the total distance the object moves:
d= v 0 t+ 1 2 a t 2 .
Deriving the position from the distance One last step remains to get the equation for position x, because the equation above gives us the total distance d moved, but does not give us position. Distance is a change in position: d = x – x0. Substituting this expression yields equation (2.6), which relates position x at any time t to initial position x0, initial velocity v0, and acceleration a.
(2.6) 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)
Position
in accelerated motion
What does the equation mean? Equation (2.6) 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 due to changes in speed that come from acceleration. Note that if the acceleration is zero, we get back equation (2.2)!
Understanding the different terms in the equation for position as a function of time
Test your knowledge A baseball is thrown straight upward and returns to the point from which it was thrown after 5 seconds. Find the baseball's original speed.
  1. 49 m/s
  2. 4.9 m/s
  3. 122.5 m/s
  4. 24.5 m/s
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