Chapter 7 - Linear Motion

7.1 Displacement, Speed and Velocity

Centre of Mass

  • Simplification of objects

  • Centre of mass or Centre of gravity (CoG)

  • The balance point

Position

  • Position - describes the location of an object at a certain point in time with respect to the origin

  • Vector quantity

Example →

Distance Travelled

  • distance travelled (d) - describes the length of the path covered during an objects entire journey

  • scalar quanity

Example → sophie swim 3 laps of a 50m pool. 50 + 50 + 50 = 150m

Displacement

  • Displacement (s) - change in position of an object given direction.

  • s = final position - initial position

  • vector quantity

  • measured in meters

Example → Sophie swam 50 meters and then swam back 30. Final position is +20. Initial is 0.     20 - 0 = +20m

  • Magnitude, units and direction are required

  • Total displacement = su of individual displacements

Speed and Velocity

  • Speed - the rate at which distance is travelled. Is scalar

  • Velocity - the rate at which displacement changes. is Vector

  • Units for both are m s-1 or km h-1

Instantaneous Speed and Velocity

  • Gives a measure of how fast something is moving at a point in time

Average Speed and Velocity

  • Gives an indication of how fast an object is moving over a time interval

average speed = distance travelled / time taken (vav= d / t)

average velocity = displacement / time taken (vav = s / t = v + u / 2)

    ⭐ ️give direction for velocity

Converting Between km h-1 and m s-1

7.2 Acceleration

  • Acceleration - measure of how quickly the velocity changes in an object.

Finding the change in Velocity and Speed

Δv = v – u

Δv → change in speed/velocity

v → final speed/velocity

u → initial speed/velocity

⭐️ Direction is required for velocity (vector)

Acceleration

  • Negative acceleration can mean an object is slowing down in direction of travel

    Also, speeding up but in opposite direction

  • Vector quantity

a  = change in velocity / time taken

    = v / t

    = (v - u) / t

7.3 Graphing Position, Velocity and Acceleration Over Time

  • When motion is 1D, info can be presented graphically

  • Nature of motion can be seen clearly

Position-Time Graphs (x-t)

  • Indicated the position of an object at any time over a period of time

  • Position, not displacement

Example →

  • For first 25 seconds, she swims at a constant rate. From 25 s to 35 s, her position doesn’t change (resting/stationary)

  • Swims in a negative direction from 35s to 60s, swimming back to starting point

  • Displacement = 25 - 40 = -15m

  • Velocity = +2 ms-1 and then 0 ms-1 and then 1 ms-1

Gradient = velocity

Positive velocity = positive direction

Negative velocity = negative direction

gradient of x-t graph = rise / run = x / t

Non-uniform Velocity

  • For motion with constant velocity, the graph will be straight (uniform)

  • For motion with different velocity, graph will be curved (non-uniform)

    • Instantaneous velocity will be the gradient of the tangent to the line at the point of interest

    • Average velocity will be the gradient of the chord between these points

Velocity-Time Graphs (v-t)

  • velocity against time shows how the velocity of an object changes with time

  • Example →

    • Moves positively at 3 ms-1 for first 4 seconds, then continues positively but slows down over the next two seconds, is stationary for 1 second, accelerates negatively for a second, then for another second at a velocity of -1ms-1 and then slows down and stops at 10 seconds.

⭐️When graph is below x axis, velocity is negative, which indicates travel in reverse direction

Finding Displacement

  • The area under a velocity-time graphs (area)

  • Area under the x axis indicates negative displacement

  • Seperate into simple equations

Rectangle = b x h

Triangle = ½ x b x h

Acceleration from a Velocity-Time Graph

  • Gradient = average acceleration

  • Gradient = rise / run

Example

Distance travelled

  • Area under

  • Regardless of direction, area of all shapes must be ADDED up

  • As seen in example above

Non-Uniform Acceleration

  • For constant acceleration, motion will be straight (uniform)

  • Non-uniform will be curved

    • Instantaneous acceleration will be gradient of the chord between two points.

    • Displacement can be calculated, but will be an estimation.

Acceleration-Time Graphs (a-t)

  • Indicates the acceleration of the object during a time period

  • Area under = velocity

  • area = a × Δt = Δv

a) is v-t, and b) is a-t

  • From 4-6s the area shows  a Δv of –3 m s−1

    • Indicates she has slowed down to 3 m s−1 during time

  • Initial speed is 3 m s−1 so she must be stationary after 6 seconds

7.4 Equations for uniform acceleration

Deriving the Equations

To logically obtain, deduce, or construct a mathematical formula from more basic principles, definitions, or known laws.

  • v = u + at 

  • s = ½ (u + v)t

  • s = ut + 1/2 at2

  • s = vt - ½ at2

  • v2 = u2 + 2as

s is the displacement (in m)

u is the initial velocity (in m s−1)

v is the final velocity (in m s−1)

a is the acceleration (in m s−2)

t is the time taken (in s).

Solving Problems Using Equations

1) Draw diagram of situation

2) Write down information given (suvat)

3) select equation that matches data

4) Use appropriate number of significant figures

5) Include units and specify direction if vector.

7.5 Vertical Motion

  • Falling objects speed up because of gravity

  • Air resistance factor

Analysing Vertical Motion

  • Some objects effected by air resistance more than others

  • If air resistance can be ignores, all bodies in free fall near the earth’s surface will move with an equal downwards acceleration

  • Mass would not happen

  • Acceleration of a free falling body is constant, it is appropriate to use same equations for uniform acceleration