Summary and Main Ideas

Motion Equations for Constant Acceleration in One Dimension

• To simplify calculations we take acceleration to be constant, so that \(\bar{a} = a\) at all times.
• We also take initial time to be zero.
• Initial position and velocity are given a subscript 0; final values have no subscript. Thus,
  

  \(\left.\begin{array}{lll}
  \Delta t & = & t \; \\
  \Delta x & = & x - x_0 \; \\
  \Delta v & = & v - v_0 \;
  \end{array}\right\}\)

• The following kinematic equations for motion with constant \(a\) are useful:
  

  \(x = x_0 + \bar{v} t\)

  \(\bar{v} = \cfrac{v_0 + v}{2}\)

  \(v = v_0 + at\)

  \(x = x_0 + v_0t + \cfrac{1}{2}at^2\)

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

• In vertical motion \(y\) is substituted for \(x\).

Problem-Solving Basics for One-Dimensional Kinematics

The six basic problem solving steps for physics are:

Step 1. Examine the situation to determine which physical principles are involved.

Step 2. Make a list of what is given or can be inferred from the problem as stated (identify the knowns).

Step 3. Identify exactly what needs to be determined in the problem (identify the unknowns).

Step 4. Find an equation or set of equations that can help you solve the problem.

Step 5. Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units.

Step 6. Check the answer to see if it is reasonable: Does it make sense?

Falling Objects

• An object in free-fall experiences constant acceleration if air resistance is negligible.
• On Earth, all free-falling objects have an acceleration due to gravity \(g\), which averages
  \(g = 9.80\text{ m/s}^2\)
• Whether the acceleration \(a\) should be taken as \(+g\) or \(-g\) is determined by your choice of coordinate system. If you choose the upward direction as positive, \(a = -g = -9.80\text{ m/s}^2\) is negative. In the opposite case, \(a = +g = 9.80\text{ m/s}^2\) is positive. Since acceleration is constant, the kinematic equations above can be applied with the appropriate \(+g\) or \(-g\) substituted for \(a\).
• For objects in free-fall, up is normally taken as positive for displacement, velocity, and acceleration.

Glossary

Free-fall

the state of movement that results from gravitational force only

Acceleration due to gravity

acceleration of an object as a result of gravity