Lecture 2 (acceleration and speed)

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Bridging Course - Lecture 2 (acceleration and speed)

The techniques of finding slopes (how quickly something changes in time or space) and finding areas (the effect of something through time or space) will now be extended to speed and acceleration.

In this lecture the following are introduced:
• Average and instantaneous acceleration
• Speed vs time graph to acceleration vs time graph
• Acceleration vs time graph to speed vs time graph
• Constant acceleration graphs and equations

Definition of average acceleration

This will have the S.I. unit of m.s^-2.

Definition of instantaneous acceleration

The instantaneous acceleration is the slope of the tangent to a speed vs time graph at a particular time.

By working out the slopes at every point on a speed vs time graph, we can construct the corresponding instantaneous acceleration vs time graph.

Instantaneous acceleration and the speed vs time graph

At t₁ the acceleration is low.
At t₂ the acceleration is large.
At t₃ the acceleration is zero and the speed is constant (but not zero).
At t₄ there is a reduction in speed.

Finding the speed vs time graph from the acceleration vs time graph.

The area under the acceleration vs time graph gives the effect of acceleration through time, i.e. it give the speed change.

Example 5
An object accelerates from zero speed at its origin and at a constant rate of 3 m.s^-2. Find the speed and position as functions of time while it continues at this rate.

A change in a quantity is signalled by a triangle Δ which is really "Delta", the capital "D" in Greek. The area under the acceleration vs time graph gives the effect of acceleration through time, which is a speed change and written as Δv.
Δv = final speed - initial speed = v - v₀ = 3t
This is the length of the vertical arrow at time t on the speed vs time graph.

The speed at any time is given by v = v₀ + 3t
but the intial speed is zero, so v = 3t
The change in position will be the effect of speed through time, i.e. the triangular area on the speed vs time graph.
Δx = final position - initial position = ½×t×3t = 1.5·t².
but the initial position is zero, so x = 1.5·t².

Position-Speed-Acceleration graphs summary.

The blue slope line on the position vs time graph gives the instantaneous speed which is the blue vertical arrow in the speed vs time graph below it.
The curvey blue line of the speed vs time graph comes from all the slopes in the graph above it.

The black slope line on the speed vs time graph gives the instantaneous acceleration which is the vertical black arrow in the acceleration vs time graph below it.
The curvey blue line of the acceleration vs time graph comes from all the slopes in the graph above it.

The green area under the acceleration vs time graph gives the change in speed, which is the green vertical arrow in the speed vs time graph above it.

The red area under the speed vs time graph gives the change in position, which is the red vertical arrow in the position vs time graph above it.

Example 6
A car is at its origin at time t=0s. It then has a speed given by v = 5 + 10t m.s^-1. Find the acceleration and position at 3s.

The speed vs time graph is:

The rectangle represents the change in position that would have occured if the car continued at its initial speed.
The triangle represents the additional distance that was added by the acceleration.
Since the car started from the origin, the final position is simply the change in position.

Constant acceleration graphs and equations

When a body has constant acceleration "a", the following applies.

Constant acceleration under gravity

The Earth's force of gravity pulls mass towards it centre. Near the earth's surface, gravity causes masses to accelerate downwards with a constant value of 9.8 m.s^-2. Using "g" as the symbol for this uniform acceleration, the following applies.

Summarising:


Instantaneous positions, speeds and accelerations.	When a body has constant acceleration "a", the following applies.

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