Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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The Acceleration Vector
When all three acceleration components are constant, i.e., when the v_{x}  t, v_{y}  t, and v_{z}  t graphs are all linear, we can define the acceleration vector as a = Δv/Δt , [only for constant acceleration] which can be written in terms of initial and final velocities as a = (v_{f}  v_{i})/Δt . [only for constant acceleration] If the acceleration is not constant, we define it as the vector made out of the a_{x}, a_{y}, and a_{z} components found by applying the slopeof thetangentline technique to the v_{x}t, v_{y} t, and v_{z} t graphs. Now there are two ways in which we could have a nonzero acceleration. Either the magnitude or the direction of the velocity vector could change. This can be visualized with arrow diagrams as shown in figures b and c. Both the magnitude and direction can change simultaneously, as when a car accelerates while turning. Only when the magnitude of the velocity changes while its direction stays constant do we have a Δv vector and an acceleration vector along the same line as the motion.
If this all seems a little strange and abstract to you, you're not alone. It doesn't mean much to most physics students the first time someone tells them that acceleration is a vector, and that the acceleration vector does not have to be in the same direction as the velocity vector. One way to understand those statements better is to imagine an object such as an air freshener or a pair of fuzzy dice hanging from the rearview mirror of a car. Such a hanging object, called a bob, constitutes an accelerometer. If you watch the bob as you accelerate from a stop light, you'll see it swing backward. The horizontal direction in which the bob tilts is opposite to the direction of the acceleration. If you apply the brakes and the car's acceleration vector points backward, the bob tilts forward. After accelerating and slowing down a few times, you think you've put your accelerometer through its paces, but then you make a right turn. Surprise! Acceleration is a vector, and needn't point in the same direction as the velocity vector. As you make a right turn, the bob swings outward, to your left. That means the car's acceleration vector is to your right, perpendicular to your velocity vector. A useful definition of an acceleration vector should relate in a systematic way to the actual physical effects produced by the acceleration, so a physically reasonable definition of the acceleration vector must allow for cases where it is not in the same direction as the motion.
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Home Newtonian Physics Vectors and Motion The Acceleration Vector 