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Acceleration
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Concept Version 8
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Motion with Constant Acceleration

Constant acceleration occurs when an object's velocity changes by an equal amount in every equal time period.

Learning Objective

  • Describe how constant acceleration affects the motion of an object


Key Points

    • Assuming acceleration to be constant does not seriously limit the situations we can study and does not degrade the accuracy of our treatment.
    • Due to the algebraic properties of constant acceleration, there are kinematic equations that can be used to calculate displacement, velocity, acceleration, and time.
    • Calculations with constant acceleration can be done in relation to one-dimensional motion as well as two-dimensional motion.

Terms

  • kinematic

    of or relating to motion or kinematics

  • acceleration

    The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).


Full Text

An object experiencing constant acceleration has a velocity that increases or decreases by an equal amount for any constant period of time. Acceleration can be derived easily from basic kinematic principles. It is defined as the first time derivative of velocity (so the second derivative of position with respect to time):

$\displaystyle a= \frac {\partial v}{\partial t}=\frac {\partial ^2 x}{\partial t^2}$

Assuming acceleration to be constant does not seriously limit the situations we can study and does not degrade the accuracy of our treatment, because in a great number of situations, acceleration is constant. When it is not, we can either consider it in separate parts of constant acceleration or use an average acceleration over a period of time.

The motion of falling objects is a simple, one-dimensional type of projectile motion in which there is no horizontal movement. For example, if you held a rock out and dropped it, the rock would fall only vertically downward toward the earth . If you were to throw the rock instead of just dropping it, it would follow a more projectile-like pattern, similar to the one a kicked ball follows.

One-Dimensional Motion

When you drop an object, it falls vertically toward the center of the earth due to the constant acceleration of gravity.

Projectile motion is the motion of an object thrown or projected into the air and is subject only to the acceleration of gravity. The object thrown is called a projectile, and the object's path is called its trajectory. In two-dimensional projectile motion, there is both a vertical and a horizontal component.

Due to the algebraic properties of constant acceleration, there are kinematic equations that relate displacement, initial velocity, final velocity, acceleration, and time. A summary of these equations is given below.

$x=x_0+ \bar{v}t$

$\displaystyle \bar{v} = \frac{v_0 +v}{2}$

$v = v_0 + at$

$\displaystyle x= x_0+v_0t+ \frac{1}{2}at^2$

$v^2 = v_0^2 + 2a(x-x_0)$

Constant Acceleration Explained with Vectors and Algebra

This video answers the question "what is acceleration? ".

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