# 1D Kinematics

## 3. One-Dimensional Kinematics

Kinematics is only the description of motion - it does not attempt to explain motion, and is thus not concerned with forces.

### 3.1 Position, path, displacement and distance

• Position and motion must be described relative to something (usually the earth).
• Position in one dimension (along a straight line, the x axis) is given by a value of x which can be positive or negative.
• Position in 2 or 3 dimensions is given by the position vector r.
• Displacement = change in position = distance moved in a given direction e.g. 2 km south or 3 m up. If the motion is in a straight line (‘one dimensional motion’) then a positive or negative sign can be used to indicate the direction of motion, provided the ‘positive’ and ‘negative’ directions have been agreed upon. For example, if the positive direction has been specified as 'up' then a position of -2 cm would mean 2cm below the zero position.
• For displacement in one dimension, displacement = change in position = Δ x = x2 - x1 where the Greek letter delta (Δ) is used to signify change.
• For displacement in 2 or 3 dimensions, displacement = change in position = Δ r = r2 - r1

### 3.2 Speed and velocity defined

• Speed is a measure of how fast something is moving.
• Speed is the rate at which distance is covered, and it is measured in units of distance divided by time (in the SI system, m/s)
• Average speed is the total distance covered divided by the time interval.
• Instantaneous speed is the speed at any instant, or the average speed over an extremely short time interval.
• Average velocity is speed together with direction, or total displacement divided by time. The total displacement is found by adding the displacement vectors using vector addition. vav = (r2 - r1) / (t2 - t1) = Δr / Δt
• Instantaneous velocity is the velocity at a given instant, or the average velocity over an extremely small time interval at that instant (the limit as the time interval approaches zero)
• In calculus notation the letter d is used to signify infinitesimal changes and therefore in calculus notation the equation for instantaneous velocity can be written : v = dr / dt
• Velocity is constant only when speed and direction are both constant.

### 3.4 Average velocity and instantaneous velocity

• Average velocity = total displacement divided by time.
• Instantaneous velocity is the velocity at a given instant, or the average velocity over an extremely small time interval at that instant.
• Thus the distinction between average velocity and instantaneous velocity is that average velocity can be calculated over any time interval, whereas instantaneous velocity must be calculated over an extremely small time interval.
• On a v-t graph, the instantaneous velocity at a given time is equal to the slope of a tangent drawn to the curve at that time.
• In calculus, this 'extremely small' time interval would be defined as the limit as the time interval approaches zero, and would be represented as dt
• Whereas the average velocity can be calculated as one value for a given journey, the instantaneous value often changes continuously.

### 3.5 Acceleration is the rate at which velocity is changing with respect to time.

• Acceleration is the rate of change of velocity. Velocity is changing when speed is changing or when direction is changing, thus an object may be accelerating even when its speed is constant.
• Since acceleration is related to changes in velocity rather than speed, acceleration is a vector quantity, like velocity itself.
• Average acceleration is defined to be the change in velocity divided by the total time taken: aav = (v2 - v1) / (t2 - t1) = Δv / Δt
• Instantaneous acceleration is the change in velocity divided by the time taken over a very short time interval (the limit as the time interval approaches zero). In calculus notation: a = dv / dt
• In the SI system, acceleration is measured in meters per second per second or (better) meters per second squared (m/s²).

### 3.6 Algebraic equations that describe uniformly accelerated straight line motion (pp. )

Assuming Δx = displacement, v = initial velocity, v0 = final velocity then

• a = (v - v0) / t
• v = v0 + a t
• Δx = ½ (v + v0) t (note that ½ (v + v0) just gives the average velocity)
• Δx = v0 t + ½ a t 2
• v2 = v02 + 2 a Δx

Note: The above equations are often called the 'equations of motion'.

### 3.8 Free fall (p. )

An object in free fall is falling under the influence of gravity alone i.e. when air resistance does not affect its motion.

• In a given location, the acceleration of all objects in free fall is the same, regardless of their mass.
• An object in free fall close to the surface of planet earth has a constant acceleration of about 10 m/s2 (9.81 m/s2 to be more precise). This acceleration, the acceleration of free fall, is known as 'g'.
• When an object falls through a fluid (e.g. air) then it does NOT experience the constant 'acceleration of free fall'. Instead, it accelerates only until it reaches its terminal speed. (Note: A fluid is anything that can flow, i.e. any gas or liquid.)
• At terminal speed in air, the force of air resistance balances the force of gravity.