Analytical toolbox studies Introductory kinematics by Drs J. Whitty & B. Henderson Learning Outcomes Use Distance-Time Graphs Use Velocity-Time Graphs Derivation of the Linear Equations of Kinematics Apply of the Equations of Kinematics Answer text book type questions on these topics

Distance-Time Graphs Consider an aeroplane travelling at 500km in the first hour of Flight and 750km in the second, plotting distance travelled against time Renders: Distance (km) time The average speed is:

Velocity-Time Graphs Consider a helicopter moves to velocity of 4m/s in seconds then increases its velocity to 8m/s in the next 3 seconds. Plotting the velocity against time Renders: Velocity (m\s) Time (s) The accelerations are:

Velocity-Time Graphs Or more generally: Velocity (m\s) Time (s) t u v The average acceleration is Where u is the initial velocity and v is the final velocity Slope

Area Under the v-t Graph Consider a body with an initial velocity u at some time t o which travels in a time t to a velocity of v Velocity (m\s) Time (s) toto t u v

The Equations of Kinematics Class Example: Derive the area of a trapezium Now consider the aforementioned Velocity-Time Graph Velocity (m\s) Time (s) toto t u v Area=The distance travelled in the time taken to change the velocity

The Equations of Kinematics We have seen that: And the area under the graph renders: Hence (1) (2) Substitution of (1) into (2) renders: Therefore: (3) Substitution of (2) into (1) for t renders: Therefore: (4)

The Equations of Kinematics Summary: (3) (4) (1) The Gradient of the v-t Graph (2) The Area Under the v-t Graph The Slope of the s-t Graph Class Problems

1. A train accelerates uniformly from rest to 60kph in 6min, after which the speed is constant, evaluate time taken to travel 6km. 2. If a train retards uniformly from 30m/s to 16m/s in 200m, evaluate the retardation and the total time of the retardation 3. A racing car engine is shut off. In the first 30s it covers 110m, and then comes to rest after a further 30s. Evaluate the initial speed and the total distance travelled.

Problem #1: solution 1. Here we have v=0, u=60kph, t=6min (implying eqn 2) as we need s. So the train has travelled half 3km, but is now at constant speed i.e. u=v=60/3.6=16.333m/s, we can use the same equation again here only need it to travel a further 3km, thus:

Problem #1: alternative solution We could obtain a direct solution from the velocity-time graph thus t-360s 360s This is the area of A triangle plus that of a rectangle 60/3.6

Problem #2: solution 2. Here v=16m/s, u=30m/s, s=200: t=? To evaluate the retardation we could simply just apply equation (1), however, we don’t have a time so in this case it is useful to use equation (2) first and then apply (1). i.e. answer the question backwards! Now apply equation (1)

Problem #3: solution 3. Here we do our detective work and find that when t=30, s=210 s: u=? and v=? Note the ‘two times’ imply the simultaneous equations

Summary Have we met our learning objectives: In particular are you able to: Use Distance-Time Graphs & Velocity-Time Graphs Derivation of the Linear Equations of Kinematics Apply of the Equations of Kinematics Apply Energy methods to solve engineering problems Answer examination type questions on these topics