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# 3: Motion Along a Straight Line

A full treatment of kinematics considers motion in two and three dimensions. For now, we discuss motion in one dimension, which provides us with the tools necessary to study multidimensional motion. A good example of an object undergoing one-dimensional motion is the maglev (magnetic levitation) train depicted at the beginning of this chapter. As it travels, say, from Tokyo to Kyoto, it is at different positions along the track at various times in its journey, and therefore has displacements, or changes in position. It also has a variety of velocities along its path and it undergoes accelerations (changes in velocity). With the skills learned in this chapter we can calculate these quantities and average velocity. All these quantities can be described using kinematics, without knowing the train’s mass or the forces involved.

• 3.1: Prelude Motion Along a Straight Line
We can describe motion using the two disciplines of kinematics and dynamics. We study dynamics, which is concerned with the causes of motion, in Newton’s Laws of Motion; but, there is much to be learned about motion without referring to what causes it, and this is the study of kinematics. Kinematics involves describing motion through properties such as position, time, velocity, and acceleration.
• 3.2: Position, Displacement, and Average Velocity
To describe the motion of an object, you must first be able to describe its position (x): where it is at any particular time. More precisely, we need to specify its position relative to a convenient frame of reference. A frame of reference is an arbitrary set of axes from which the position and motion of an object are described.
• 3.3: Instantaneous Velocity and Speed
The quantity that tells us how fast an object is moving anywhere along its path is the instantaneous velocity, usually called simply velocity. It is the average velocity between two points on the path in the limit that the time (and therefore the displacement) between the two points approaches zero.
• 3.4: Average and Instantaneous Acceleration
Acceleration is the rate at which velocity changes. It is also a vector, meaning that it has both a magnitude and direction. The SI unit for acceleration is meters per second squared. Acceleration can be caused by a change in the magnitude or the direction of the velocity, or both. Instantaneous acceleration is the slope of the velocity-versus-time graph.
• 3.5: Motion with Constant Acceleration (Part 1)
When analyzing one-dimensional motion with constant acceleration, identify the known quantities and choose the appropriate equations to solve for the unknowns. Either one or two of the kinematic equations are needed to solve for the unknowns, depending on the known and unknown quantities.
• 3.6: Motion with Constant Acceleration (Part 2)
Two-body pursuit problems always require two equations to be solved simultaneously for the unknowns.
• 3.7: Free Fall
An object in free fall experiences constant acceleration if air resistance is negligible. On Earth, all free-falling objects have an acceleration g due to gravity, which averages g = 9.81 m/s^2. For objects in free fall, the upward direction is normally taken as positive for displacement, velocity, and acceleration.
• 3.8: Finding Velocity and Displacement from Acceleration
Integral calculus gives us a more complete formulation of kinematics. If acceleration a(t) is known, we can use integral calculus to derive expressions for velocity v(t) and position x(t).
• 3.E: Motion Along a Straight Line (Exercises)
• 3.S: Motion Along a Straight Line (Summary)

Thumbnail: A JR Central L0 series five-car maglev (magnetic levitation) train undergoing a test run on the Yamanashi Test Track. The maglev train’s motion can be described using kinematics, the subject of this chapter. (credit: modification of work by “Maryland GovPics”/Flickr).