doPhysics · Kinematics
Motion in One Dimension
Motion along a straight line. Position, distance and displacement, speed and velocity, acceleration, the three equations of motion, and free fall under gravity.
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Why this chapter
Why this chapter exists
Every chapter of mechanics asks the same three questions about a body. Where is the body? How fast is the body moving? How is that speed changing? This chapter answers the three questions for the simplest case, motion along a straight line. When curved paths and forces arrive in later chapters, the words position, velocity and acceleration will already be yours.
You will learn
By the end of this chapter you will
- Say, in one sentence each, how distance differs from displacement and how speed differs from velocity.
- Read position, velocity and acceleration from a graph by height, slope and area.
- Derive and use the three equations of motion, and state the one condition under which they hold.
- Solve free-fall problems with one sign convention, using symmetry instead of memorised cases.
Before you begin
Before you begin
You should be able to add and subtract signed numbers, read a value from a graph, and solve a quadratic equation. You should know that a derivative is a rate of change. Vectors are helpful but are not needed yet, because a straight line has only two directions.
Lessons
1
Rest and Motion
What kinematics describes, how position is measured from an origin in a frame of reference, why rest and motion are relative, and when a body may be treated as a point.
Kinematics describes motionPosition in a frame of referenceRest and motion are relativePoint particle model
2
Distance and Speed
Distance is the length of the actual path. Average speed is total distance divided by total time. The two standard cases, equal times and equal distances, and why the second gives a smaller answer.
Distance is path lengthAverage speed is total distance over total timeEqual times give the arithmetic meanEqual distances give the harmonic mean
3
Displacement and Velocity
Displacement is the change in position, a vector that ignores the path. Average velocity is displacement divided by time. Why average speed and the magnitude of average velocity differ.
Displacement is a vectorAverage velocity is displacement over timeSpeed versus velocity
4
Instantaneous Speed and Velocity
From the average over an interval to the value at one instant. The limit, the derivative, and the tangent to the position-time graph.
Instantaneous velocity is a limitVelocity is the slope of the tangentInstantaneous speed is the magnitude of velocity
5
Acceleration
Acceleration is the rate of change of velocity. Average and instantaneous acceleration, the second derivative of position, and the rule that decides whether a body speeds up or slows down.
Acceleration is the rate of change of velocityInstantaneous acceleration is a derivativeSign of acceleration
6
Equations of Motion
The three equations for motion with constant acceleration, where each one comes from, which one to use, and the single condition under which all three hold.
Equations hold for uniform accelerationFirst equation from constant rateSecond equation as areaThird equation eliminates time
7
Free Fall
Motion under gravity alone. The same acceleration for every body, one sign convention for every problem, and the symmetry of rise and fall that removes the need to memorise cases.
Free fall is motion under gravity aloneOne sign convention for free fallRise and fall are symmetric
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- acceleration
- The rate of change of velocity with time, \(a = dv/dt\). A body speeds up when \(v\) and \(a\) have the same sign.
- average speed
- Total distance divided by total time, including any time spent at rest.
- average velocity
- Displacement divided by the time interval. A vector; on a line it carries a sign.
- chord
- A straight line joining two points on a curve. On a position-time graph its slope is the average velocity between those two times.
- derivative
- The rate of change of one quantity with respect to another, obtained as a limit of a ratio of small changes.
- displacement
- The change in position, a vector from the initial position to the final position. It ignores the path.
- distance
- The total length of the actual path covered by a body. A scalar that never decreases during motion.
- equations of motion
- The three relations between \(u\), \(v\), \(a\), \(t\) and \(s\) that hold while the acceleration is constant.
- frame of reference
- A coordinate system attached to an observer, together with a clock. Every position and every time is measured in some frame of reference.
- free fall
- Motion in which gravity is the only force acting on the body. Air resistance is neglected.
- g
- The magnitude of the acceleration due to gravity near the surface of the Earth, about \(9.8\,\text{m/s}^2\), often taken as \(10\,\text{m/s}^2\).
- instantaneous
- Belonging to one instant of time, not to an interval. The instantaneous velocity is the velocity at one moment.
- kinematics
- The part of mechanics that describes motion using position, velocity and acceleration, without asking which force causes the motion.
- limit
- The value that an expression approaches as a variable is made closer and closer to some number, without ever reaching it.
- observer
- The person, or the instrument, that measures positions and times. Every observer carries a frame of reference.
- origin
- The fixed point from which positions are measured. The origin has position zero.
- point particle
- A body treated as a single point that has mass but no size. The model is valid when the size of the body is very small compared with the distances involved.
- position
- The location of a body measured from the origin. On a straight line, one signed number.
- scalar
- A quantity that is added by ordinary arithmetic, with no regard to direction. Distance, speed, mass, time and energy are scalars. Electric current has a direction along a wire, but currents add as plain numbers, so current is a scalar.
- sign convention
- The choice of which direction along the line is positive. Every displacement, velocity and acceleration then carries a sign relative to that choice.
- tangent
- The straight line that touches a curve at one point and has the same slope as the curve there. On a position-time graph its slope is the velocity.
- uniform acceleration
- An acceleration that stays the same in magnitude and in direction throughout the motion.
- vector
- A quantity that needs its direction to be added. Two vectors are added by the triangle law, not by plain arithmetic. Displacement, velocity, acceleration and force are vectors.
- velocity
- The rate of change of position with time, \(v = dx/dt\). Its magnitude is the speed and its sign is the direction.