Speed
Introduction

Mass, length, and time are three of the most fundamental physical quantities. They are called base quantities. You will already have met their units – kilograms, metres, and seconds. The sizes of 1 kg, 1 m, and 1 s are set by international definition.

Knowing the size of physical quantities is important for standardizing measurements in different areas. The standard kilogram is a physical object (see above), while the standard second is defined in terms of other scientific phenomena.
Base and derived quantities
Mass, length, and time are called base quantities. They are just three of the possible base quantities used in the SI (Système International) system of measurement.
 Base quantity Name Symbol Mass kilogram kg Length metre m Time second s Electric current ampere A Thermodynamic temperature kelvin K Amount of substance mole mol Luminous intensity candela cd

Figure .  List of base quantities.

Derived quantities are defined as combinations of the base quantities listed above. For example, speed and
velocity
An object's velocity states both the speed and direction of motion relative to a fixed reference point.
velocity
are derived quantities and are both combinations of length and time.

Acceleration
An object's acceleration is its rate of change of velocity.
Acceleration
is a …
Acceleration is defined in terms of which of the following base quantities?
•  Mass No Yes Length No Yes Electric current No Yes Time No Yes

The units for derived quantities are combinations of the units of the corresponding base quantities.

 Derived quantity Units Speed ms−1 Acceleration ms−2

Figure .  Some simple derived quantities.

Sometimes the size of derived quantities is stated in non-standard units, but this is equally acceptable as there will always be derived equivalents.

 Derived quantity Unit name Specific unit Equivalent unit Frequency hertz Hz s−1 Force newton N kgms−2 Pressure pascal Pa kgm−1s−2 Energy joule J kgm2s−2

Figure .  List of derived quantities with specific and equivalent units.

Momentum
The momentum of a moving object is the product of its mass and velocity.
Momentum
is defined as the product of mass and velocity. Is momentum a base quantity?
The units of momentum will involve which of the following base units?
•  Mass No Yes Length No Yes Electric current No Yes Time No Yes
Describing motion
Match the following events with the standards achieved by the gold medallists in the 2000 Olympics.
•  Men's 100 m 21.64 s 260 kg 6.99 m 68.91 m 69.30 m 9.87 s Women's 200 m 21.64 s 260 kg 6.99 m 68.91 m 69.30 m 9.87 s Superheavyweight lifting clean and jerk 21.64 s 260 kg 6.99 m 68.91 m 69.30 m 9.87 s Women's long jump 21.64 s 260 kg 6.99 m 68.91 m 69.30 m 9.87 s Men's discus 21.64 s 260 kg 6.99 m 68.91 m 69.30 m 9.87 s Women's javelin 21.64 s 260 kg 6.99 m 68.91 m 69.30 m 9.87 s

It is easy to match the weightlifting record because there is only one answer in kilograms. Matching the 100 m and 200 m times is also quite straightforward, as we would naturally expect the time for the 200 m to be larger because the distance is greater. Matching the javelin and the discus distances is not quite so easy as their magnitudes are similar.

Humans have a natural perception of time, distance, and mass because we relate new circumstances to our existing experience. When quantities are of a similar size scientific methods involving accurate measurements are needed to distinguish between them.

To compare how quickly objects are moving we can determine their average speeds. Average speed is defined as

The blue and green balls in Fig.4 have similar speeds. To be certain which is moving faster, you will need to measure the time taken to complete a certain distance and then calculate their average speeds. The stopwatch in Fig.4 can be used for this purpose.

 Figure 4. Determining speed.
Complete the following statements to determine the speeds of the blue and green balls.

• The time taken by the blue ball to travel 6 m is approximately seconds.

The time taken by the green ball to travel 8 m is approximately seconds.

The time taken by the blue ball to travel 10 m is approximately seconds.

The time taken by the green ball to travel 10 m is approximately seconds.

Simply comparing the times in the previous question is not sufficient to decide which is moving faster, the distance travelled must also be taken into account. However, we can use the definition of average speed to make a direct comparison.

Complete the following statements to calculate the average speeds of the blue and green balls.

• The average speed of the blue ball during the entire 6 m of its journey is  ms−1.

The average speed of the green ball during its 10 m journey is  ms−1.
Which ball is moving faster?
Measuring average speeds
Finding an average speed involves accurately measuring the time to travel a known distance. Measuring distances with a ruler is sufficiently accurate if the distance is more than a few centimetres, but measuring short time intervals accurately by manually starting and stopping a stopwatch is more difficult. To overcome this problem you can use a system of light gates like that shown in Fig.5.

• When the card on the trolley interrupts the light beam at A the electronic timing system starts.

• Timing continues until the card interrupts the light beam at B.

• The clock within the computer measures the time interval between the start and stop signals very accurately.

 Figure 5. Determining travel times with light gates.

Repeat the experiment shown in Fig.5 and record the time the trolley takes to travel from A to B on each occasion in the following table.

 Expt. no. Time to travel from A to B / s 1 2 3 4 5

To determine the average speed of the trolley as it travels from A to B in each experiment you would measure the distance between A and B and calculate the average speed using the equation,

During one experiment with the apparatus shown in Fig.5, A and B are 0.8 m apart. The time to travel between these points is measured as 0.362 seconds. What is the average speed of the trolley?
During one experiment with the apparatus shown in Fig.5, A and B are 1.20 m apart. The calculated average speed for this journey is 2.52 ms−1. What is the time taken for the trolley to complete this journey?
Instantaneous speed
You will have noticed that the trolley in Fig.5 gets faster as it rolls down the slope. The trolley is accelerating and travels faster past B than past A. The method outlined above determines the average speed between A and B.

The speed of a rolling trolley can be measured at different places along the runway with a single light gate as shown in Fig.6.

Start the experiment and note the following points:

• When the front edge of card on the trolley interrupts the light beam the electronic timing system starts.

• Timing continues while the card is interrupting the beam.

• Timing stops when the beam is reformed.

 Figure 6. Determining speeds with a single light gate.

The clock within the computer measures the time taken for the card to pass through the light beam. During this measured time interval the trolley travels a distance equal to the length of the card. With these quantities we can calculate the average speed during this short time interval.

If the time interval is short the calculated average speed will be close to the speed of the trolley at the instant timing began. The instantaneous speed of a moving object is defined as the average speed over a very short time interval.

In Fig.6 you can move the light gate by clicking on the green spot and dragging it to another position. Measure the interrupt times for the light gate positioned at each of the lines A to E. Record your measurements in the table below.

 Position of light gate First attempt interrupt time / s Second attempt interrupt time / s Third attempt interrupt time / s A B C D E

During one experiment with the apparatus shown in Fig.6 the card on the trolley is 10 cm long and takes 0.108 s to pass through the light gate. What is the speed of the trolley?
During one experiment with the apparatus shown in Fig.6 the card on the trolley is 10 cm long and the calculated speed is 2.27 ms−1. What is the measured interrupt time on this occasion?

The speed of a lorry is determined from the time taken for the rotation of one of its wheels. The distance travelled during each rotation is equal to the wheel's circumference so even at moderately low speeds the time for each revolution is short. Speedometers are widely regarded as showing instantaneous speeds.

 Figure 7. A tachograph.

A tachograph is a record of the instantaneous speed of a vehicle over a period of time. The instantaneous speed varies but if the total distance travelled during a 24 hr period is known the average speed can be calculated.

Complete the following statements.

• Long distance haulage businesses will use information on speeds when determining delivery times but the driver and local police forces will be more concerned with speeds.

 Figure 8. Speed limit.
Speed cameras and other devices measure instantaneous speeds, although some also measure average speed over a period of time as a method of confirming the accuracy of the instantaneous measurement.

The World Motor Sport Council verified the world land speed records set by the Thrust supersonic car on 15 October 1997 at Black Rock Desert, Nevada (USA), as 763.035 mph (341.11 ms-1) for the 'flying mile'. This record involved measuring the time taken to travel through the stated mile and determining the average speed. A second run in the opposite direction was also required to allow for any tail winds. The stated speed is the mean of both average speeds.
 Figure 9. Now that's quick!

Summary

Speed is a derived quantity which is defined in terms of the base quantities distance and time.

The average speed of a moving object is defined as

In laboratory experiments average speeds are determined using either one or two light gates.

The average speed over a very short time interval is called the instantaneous speed.

Tachograph charts record instantaneous speeds over extended periods.

Exercises
1. During a 1500 m race an athlete completes the first 800 m in 2 minutes and 10 seconds. Calculate her average speed for 800 m.
•  ms−1   (to 2 d.p.)
2. She completes the race in 4 minutes. How long does it take her to run the remainder of the race?
• minute(s) and second(s)
3. Calculate her average speed for the remainder of the race.
•  ms−1   (to 2 d.p.)
4. Decide whether her average speed for the final 700 m is greater or less than her average speed for the first 800 m.
5. Calculate her average speed for the entire race.
•  ms−1   (to 2 d.p.)
6. A sports scientist tries to model the motion of a pitch in a game of softball. She assumes that the pitcher delivers a fast ball at a speed of 30 ms−1 and that the ball travels horizontally throughout its flight. The batter stands 15 m from the pitcher.

• How long does it take the ball to travel from the pitcher to the batter?
s   (to 1 d.p.)

It takes the batter 0.22 s to react to the pitcher's throw. How long does the batter have to swing the bat to hit the ball?
s   (to 2 d.p.)

The swinging bat travels 1.4 m before hitting the ball. Calculate the speed with which the bat hits the ball.
ms−1   (to the nearest whole number)
7. A van driver makes an average of 8 deliveries per day. In a normal working week of 5 days the driver covers a total distance of 800 km.

• Calculate the average distance travelled for each delivery.
km   (to the nearest whole number)

The fuel consumption of the van is 7.5 km per litre. Calculate the volume of fuel used by the van in a week.
litres   (to 1 d.p.)
8. Which of the following has the highest average speed?
 Figure 10. Microlight aircraft.
9. A microlight aircraft is 50 m vertically above a marker point as it attempts to land as in Fig.10. The marker point is 120 m horizontally from this landing area. What is the minimum distance the microlight has to travel before reaching the landing area?
•  m   (to the nearest whole number)
Well done!
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