8.1: One-Dimensional Equations of Motion

Last updated
Save as PDF

Page ID: 50602

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Kinetics is the branch of dynamics that deals with the relationship between motion and the forces that cause that motion. The basis for all of kinetics is Newton's Second Law, which relates forces and accelerations for a given body. In its basic form, Newton's Second Law states that the sum of the forces on a body will be equal to the mass of that body times the rate of acceleration. For bodies in motion, we can write this relationship out as the equation of motion.

\[ \sum \vec{F} = m * \vec{a} \]

In cases where accelerations only exist in a single dimension, we can reduce the above vector equation into a single scalar equation. Calling that single direction the \(x\) direction, we arrive at the single equation of motion shown below. By entering known forces or accelerations, we can use this equation to solve for a single unknown force or acceleration term.

A box of mass m is pushed along a flat, frictionless surface in the rightwards (positive x) direction by a force F_x. It experiences a rightwards acceleration of a_x. In the vertical direction, the box experiences an upwards normal force from the surface and a downwards gravitational force, producing no net movement in the y-direction. — Figure \(\PageIndex{1}\): This box being pushed along a frictionless surface can be examined as a one dimensional kinetics problem. Acceleration exists only in the \(x\) direction, related by the equation of motion to the single unbalanced force in the \(x\) direction. Because the forces in the \(y\) direction are balanced, the acceleration in that direction will be zero.

\[ \sum F_x = m * a_x = m * \ddot{x} \]

Kinetics and the equation(s) of motion relate forces and accelerations, and are often used in conjunction with the kinematics equations, which relate positions, velocities and accelerations as discussed in the previous chapter. Depending on the problem being examined, the kinematics equations may need to be examined either before or after the kinetics equations.

Video lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/CEL2cpvdQmQ.

Example \(\PageIndex{1}\)

A block with a weight of 90 pounds sits on a frictionless surface and a 50-pound force is applied in the \(x\) direction, as shown below.

What is the rate of acceleration of the block?
What is the velocity and displacement three seconds after the force is applied?

A 90-pound box sits on a flat horizontal surface, with the positive x-direction being to the right and the positive y-direction being upwards. The box experiences a pushing force of 50 lbs in the positive x-direction. — Figure \(\PageIndex{2}\): problem diagram for Example \(\PageIndex{1}\). A box on a flat, frictionless surface experiences a pushing force towards the right.

Solution: Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Majid Chatsaz. YouTube source: https://youtu.be/NwmVFrTGd0A.

Example \(\PageIndex{2}\)

A block with a weight of 90 pounds sits on a surface with a kinetic coefficient of friction of 0.2, and a 50-pound force is applied in the \(x\) direction as shown below.

What is the rate of acceleration of the block?
What is the velocity and displacement three seconds after the force is applied?

Solution: Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Majid Chatsaz. YouTube source: https://youtu.be/9BgdnzTUO9I.

Example \(\PageIndex{3}\)

A 2000-lb elevator decelerates downward, going from a speed of 25 ft/s to a stop in a distance of 50 ft.

What is the average rate of deceleration?
What is the tension in the cable supporting the elevator during this period?

A glass-sided, hemicylindrical elevator in an exterior shaft. — Figure \(\PageIndex{4}\): A descending glass-sided elevator.

Solution: Video \(\PageIndex{4}\): Worked solution to example problem \(\PageIndex{3}\), provided by Dr. Majid Chatsaz. YouTube source: https://youtu.be/I2j0peOLzro.

Example \(\PageIndex{4}\)

A rocket test sled is being used to test a solid rocket booster (mass = 1000 kg). It’s known that generally a solid rocket booster’s force will fit the equation \(F = A + Bt – C t^2\). If the rocket has an initial thrust of 10 kN, and achieves a speed of 150 m/s and travels 700 meters during a 10-second test run, determine the constants \(A\), \(B\) and \(C\) for the rocket.

A rocket sled on a track holds a solid rocket booster, moving towards the right. — Figure \(\PageIndex{5}\): A rocket sled holding a solid booster, moving rightwards on a straight track.

Solution: Video \(\PageIndex{5}\): Worked solution to example problem \(\PageIndex{4}\), provided by Dr. Majid Chatsaz. YouTube source: https://youtu.be/tmUOZXuAzNE.