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## Speed and velocity

Definition 1: Average speed

Average speed is the distance ($D$) travelled divided by the time ($Δt$) taken for the journey.

Quantity: average speed ($v av$)         Unit name: metre per second         Unit symbol: m·s−1

Definition 2: Average velocity

Average velocity is the change in position of a body divided by the time it took for the displacement to occur.

Quantity: average velocity ($v → av$)         Unit name: metre per second         Unit symbol: m·s−1

Before moving on review the difference between distance and displacement. Sometimes the average speed can be a very big number while the average velocity is zero.

Distance and displacement review

Average velocity is the rate of change of position. It tells us how much an object's position changes per unit of time. Velocity is a vector. We use the symbol $v → av$ for average velocity. If we have a displacement of $Δx →$ and a time taken of $Δt$, $v → av$ is then defined as:

$averagevelocity(inm·s -1 )=changeinposition(inm) changeintime(ins)v → av =Δx → Δt$(1)

Velocity can be positive or negative. A positive velocity points in the direction you chose as positive in your coordinate system. A negative velocity points in the direction opposite to the positive direction.

Average speed (symbol $v av$) is the distance travelled ($D$) divided by the time taken ($Δt$) for the journey. Distance and time are scalars and therefore speed will also be a scalar. Speed is calculated as follows:

$averagespeed(inm·s -1 )=distance(inm) time(ins)$(2)
$v av =D Δt$(3)

## Example 1: Average speed and average velocity

### Question

James walks 2 km away from home in 30 minutes. He then turns around and walks back home along the same path, also in 30 minutes. Calculate James' average speed and average velocity.

### Answer

#### Identify what information is given and what is asked for

The question explicitly gives

• the distance and time out (2 km in 30 minutes)

• the distance and time back (2 km in 30 minutes)

#### Check that all units are SI units.

The information is not in SI units and must therefore be converted.

To convert km to m, we know that:

$1km=1000m∴2km=2000m(multiplybothsidesby2.)$(4)

Similarly, to convert 30 minutes to seconds,

$1min=60s∴30min=1800s(multiplybothsidesby30)$(5)

#### Determine James' displacement and distance.

James started at home and returned home, so his displacement is 0 km.

$Δx →=0m$

James walked a total distance of 4000 m (2000 m out and 2000 m back).

$D=4000m$

#### Determine his total time.

James took 1800 s to walk out and 1800 s to walk back.

$Δt=3600s$

#### Determine his average speed

$v av =D Δt=4000m 3600s=1,11m·s -1$(6)

#### Determine his average velocity

$v → av =Δx → Δt=0m 3600s=0m·s -1$(7)

## Differences between speed and velocity

The differences between speed and velocity can be summarised as:

 Speed Velocity 1. depends on the path taken 1. independent of path taken 2. always positive 2. can be positive or negative 3. is a scalar 3. is a vector 4. no dependence on direction and so is only positive 4. direction can be determined from the sign convention used (i.e. positive or negative)

Additionally, an object that makes a round trip, i.e. travels away from its starting point and then returns to the same point has zero velocity but travels at a non-zero speed.

### Exercise 1: Displacement and related quantities

Bongani has to walk to the shop to buy some milk. After walking 100 m, he realises that he does not have enough money, and goes back home. If it took him two minutes to leave and come back, calculate the following:

1. How long was he out of the house (the time interval $Δt$ in seconds)?

2. How far did he walk (distance ($D$))?

3. What was his displacement ($Δx →$)?

4. What was his average velocity (in m·s−1)?

5. What was his average speed (in m·s−1)?

a) 120 seconds.

b) 200 m.

c) 0

d) The velocity is the total displacement over the total time and so the velocity is 0.

$v=\frac{0}{60}=0m\cdot {s}^{-1}$

e) The speed is the total distance traveled divided by the total time taken:

$speed=\frac{200}{120}=1,67m\cdot {s}^{-1}$

Bridget is watching a straight stretch of road from her classroom window. She can see two poles which she earlier measured to be 50 m apart. Using her stopwatch, Bridget notices that it takes 3 s for most cars to travel from the one pole to the other.

1. Using the equation for velocity ($v → av$ = $Δx → Δt$), show all the working needed to calculate the velocity of a car travelling from the left to the right.

2. If Bridget measures the velocity of a red Golf to be −16,67 m·s−1, in which direction was the Golf travelling? Bridget leaves her stopwatch running, and notices that at $t=5,0s$, a taxi passes the left pole at the same time as a bus passes the right pole. At time $t=7,5s$ the taxi passes the right pole. At time $t=9,0s$, the bus passes the left pole.

3. How long did it take the taxi and the bus to travel the distance between the poles? (Calculate the time interval ($Δt$) for both the taxi and the bus).

4. What was the average velocity of the taxi and the bus?

5. What was the average speed of the taxi and the bus?

6. What was the average speed of taxi and the bus in km·h−1?

a) We choose a frame of reference. E.g. from the left pole to the right pole is the positive direction.

The displacement ($\Delta x$ ) for a car is 50 m and the time taken ($\Delta t$ ) is 3 s.

Then the velocity for a car traveling from left to right is:

$v=\frac{\Delta x}{\Delta t}=\frac{50}{3}=16,67m\cdot {s}^{-1}$

b) The direction depends on which convention was taken as the positive direction. If the direction from the left pole to the right pole is taken as the positive direction then the car is traveling from the right pole to the left pole.

c) Taxi: $\Delta t={t}_{f}-{t}_{i}=7,5-5=2,5s$

Bus: $\Delta t={t}_{f}-{t}_{i}=9-5=4s$

d) Taxi: $v=\frac{\Delta x}{\Delta t}=\frac{50}{2},5=20m\cdot {s}^{-1}$

Bus: $v=\frac{\Delta x}{\Delta t}=\frac{-50}{4}=-12,5m\cdot {s}^{-1}$

e) Taxi: $20m\cdot {s}^{-1}$

Bus: $12,5m\cdot {s}^{-1}$

f) Taxi: $s=\frac{20m}{1s}×\frac{1km}{1000m}×\frac{60s}{1min}×\frac{60min}{1hr}=72km\cdot h{r}^{-1}$

Bus:  $s=\frac{12,5m}{1sec\right)×\frac{1km}{1000m}×\left(60\frac{sec}{1min}×\frac{60min}{1hr}=45km\cdot h{r}^{-1}}$

A rabbit runs across a freeway. There is a car, 100 m away travelling towards the rabbit.

1. If the car is travelling at 120 km·h−1, what is the car's speed in m·s−1.

2. How long will it take the a car to travel 100 m?

3. If the rabbit is running at 10 km·h−1, what is its speed in m·s−1?

4. If the freeway has 3 lanes, and each lane is 3 m wide, how long will it take for the rabbit to cross all three lanes?

5. If the car is travelling in the furthermost lane from the rabbit, will the rabbit be able to cross all 3 lanes of the freeway safely?

a) 1 km = 1 000 m and 1 hour = 60 seconds.

$\frac{120km}{1hour}×\frac{1000m}{1km}×\frac{1hr}{60min}×\frac{1min}{60sec\right)=33,33m\cdot {s}^{-1}}$

b) $\frac{100m}{33,33m\cdot {s}^{-1}}=3,00s$

c) $\frac{10km}{1hour}×\frac{1000m}{1km}×\frac{1hr}{60min}×\frac{1min}{60sec\right)=\frac{10000}{3600}=2,78m\cdot {s}^{-1}}$

d) The rabbit has to cover a total distance of 9 m. (3 times 3)

The time it will take the rabbit is:

$\frac{9m}{2,78m\cdot {s}^{-1}}=3,24s$

e) Although the rabbit will take slightly longer than the car to cross the freeway, the rabbit will reach the far lane in$\frac{6}{2},78=2,14s$ and so in the time it takes to cross the final lane the car will hit the rabbit.

### Investigation 1: An exercise in safety

Divide into groups of 4 and perform the following investigation. Each group will be performing the same investigation, but the aim for each group will be different.

1. Choose an aim for your investigation from the following list and formulate a hypothesis:

• Do cars travel at the correct speed limit?

• Is is safe to cross the road outside of a pedestrian crossing?

• Does the colour of your car determine the speed you are travelling at?

• Any other relevant question that you would like to investigate.

2. On a road that you often cross, measure out 50 m along a straight section, far away from traffic lights or intersections.

3. Use a stopwatch to record the time each of 20 cars take to travel the 50 m section you measured.

4. Design a table to represent your results. Use the results to answer the question posed in the aim of the investigation. You might need to do some more measurements for your investigation. Plan in your group what else needs to be done.

5. Complete any additional measurements and write up your investigation under the following headings:

• Aim and Hypothesis

• Apparatus

• Method

• Results

• Discussion

• Conclusion

6. Answer the following questions:

1. How many cars took less than 3 s to travel 50 m?

2. What was the shortest time a car took to travel 50 m?

3. What was the average time taken by the 20 cars?

4. What was the average speed of the 20 cars?

5. Convert the average speed to km·h−1.