1. Speed and velocity - the relationship between distance and time
Formula for speed, calculations and interpreting distance time graphs
Doc Brown's Physics Revision Notes
Suitable for GCSE/IGCSE Physics/Science courses or their equivalent
This page will help you answer questions like e.g.
What do we mean by displacement?
How do you calculate speed or velocity?
What is the difference between scalar and vector quantities in motion?
What js the difference and similarity between speed and velocity?
How do you interpret speed/velocity versus time graphs?
A few technical terms explained
A scalar quantity only has a magnitude and no specified direction
A vector quantity has both magnitude and specified direction
Distance: Distance is how far an object has moved in any direction, the direction isn't specified, so its a scalar measurement.
Displacement: Displacement is how far an object has been moved in a straight line from specified starting point to a specified finishing point, but in particular specified direction/directions, so its a vector measurement. e.g.
Speed: Speed is how fast an object is moving but no direction is specified, so its a scalar measurement.
Velocity: Velocity is how fast an object is travelling in a particular direction, so its a vector measurement.
Speed, distance and time calculations - problem solving
The speed (or velocity) of a moving object is rarely constant.
When you walk, run or travel in a car or train your/their speed is constantly changing.
The speed that a person can walk, run or cycle depends on many factors including: age, distance travelled, fitness and terrain.
Typical values may be taken as: (maths reference to put speeds in perspective: 1 hour ≡ 3600 s, 1 mile = 1.61 km)
A train travelling at 200 km/hour (~ 124 mph) is moving at a speed of ~56 m/s.
Sound travels at ~340 m/s (1224 km/hour, ~761 mph), but this varies with air pressure (density) and temperature.
The ultimate speed of anything is that of 'light' photons, all electromagnetic radiation travels at the 'speed of light' which is greatest in vacuum and is ~3.0 x 105 km/s (~3.0 x 108 m/s, 186,000 miles/s), .
The formula for speed/velocity and example calculations - problem solving
speed (metres/second, m/s) = distance travelled (metres, m) ÷ time (seconds, s)
distance (m) = average speed (m/s) x time (s)
time (s) = distance (m) ÷ average speed (m/s)
If the speed of an object is variable, average speed is simply the total distance travelled divided by the total time taken.
e.g. a sprinter completing a 200 m race in 25 seconds has an average speed of 200 / 25 = 8 m/s, but quite plainly the speed is variable as the athlete starts from 0 m/s to perhaps a maximum speed of over10 m/s since 100 m sprinters complete their distance in ~10 s giving an average speed of ~10 m/s.
For v to be a velocity, then direction of motion should be specified.
Q1.1 A train was timed to take 2.5 seconds when passing between two posts 100 m apart.
Q1.2 How far will a car travel in 15 seconds at a speed of 20 m/s?
Q1.3 A sprinter runs 400 m at an average speed of 8.4 m/s. To the nearest 0.1 s, how long did the sprint run take?
Drawing and interpreting distance - time graphs
The gradient (slope) at any point on the graph gives you the speed at that point.
Since speed = distance ÷ time, then speed = (change in vertical y axis) ÷ (change in horizontal x axis)
The steeper the gradient the greater the speed
(1) Distance - time graphs - acceleration - speeding up
Graph curves upwards, showing increasing speed/velocity with time (acceleration), for each incremental time unit (e.g. minute or second) there is an ever increasing (larger) distance covered in the same time.
(2) Distance - time graphs - motionless
Graph is flat/horizontal, indicating zero speed/velocity, there is no increase in distance with time, object has stopped moving.
(3) Distance - time graphs - deceleration - slowing down
Graph shows the curve is levelling off, steadily decreasing speed/velocity with time (deceleration), for each incremental time unit (e.g. minute or second) there is an ever decreasing (smaller) distance covered in the same time.
(4) Distance - time graphs
Graph is linear, showing constant speed/velocity, the distance covered in any equal time increment is the same.
The four graphs above illustrate of what you might see at any point on a distance-time graph, BUT, in reality, any distance-time graph will be more complicated than any of these specific graphs. So, you find a graph with all four types of gradient in just one distance-time graph - see the graph based questions below.
Examples of distance-time graph questions
You get the speed from the gradient at any given point on the graph, which is easy if the graph at that point is linear (constant speed).
For example look at the linear portion between the two purple 'blobs' on the right-hand graph.
If the graph is curved you will need to draw a tangent to the curve at the point where you want to know the speed.
For example I've drawn a tangent at ~56 seconds and converted it to a purple triangle so you can measure the gradient - see Q2.1 (c) below.
Q2.1 The graph below shows part of a car journey.
Distance - time graph for a short car journey
Interpretation question using speed = distance / time
(a) Including calculating the average speed, describe and explain the motion of the car between:
(b) What is the average speed of the car while it is moving?
(c) How can you find the specific speed of the car at any point on the graph?
Q2.2 The graph below summarises part of the journey of a train (somewhat delayed at some point!).
Interpretation question using speed = distance / time and giving your answers in km/hour and m/s
(a) What is the average speed between 1300 and 1400 hours?
(b) What is the average speed between 1530 and 1800 hours?
(c) How long was the train stopped for?
(d) What was the average speed for the whole journey?
Velocity - Time graphs and acceleration are dealt with in detail in section 2
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