Worksheet: Converting Units Using Dimensional Analysis

In this worksheet, we will practice how to use dimensional analysis to convert between different units of the same quantity to solve real-world problems.

Q1:

If 1 foot is about 30 centimeters and 1 yard is about 90 centimeters, approximately how many feet are there in a yard?

Q2:

Which quantities could be measured to determine the fuel efficiency of a car?

  • Athe distance traveled and the corresponding volume of fuel used
  • Bthe distance traveled over a given period
  • Cthe quantity of fuel used
  • Dthe fuel price and the distance traveled

Which of the following is an appropriate unit for the fuel efficiency?

  • Amiles per hour
  • Bcubic centimeters per litre
  • Cmiles per gallon
  • Dkilometers per hour
  • Ecubic centimeters per gallon

Q3:

Elizabeth wants to convert 4,500 kg/m3 into grams per cubic centimeter.

Which of the following is correct?

  • A 4 , 5 0 0 1 = 4 , 5 0 0 1 × 1 1 0 0 × 1 , 0 0 0 1 k g m k g m m c m g k g
  • B 4 , 5 0 0 1 = 4 , 5 0 0 1 × ( 1 0 0 ) 1 × 1 1 , 0 0 0 k g m k g m c m m k g g
  • C 4 , 5 0 0 1 = 4 , 5 0 0 1 × 1 ( 1 0 0 ) × 1 , 0 0 0 1 k g m k g m m c m g k g
  • D 4 , 5 0 0 1 = 4 , 5 0 0 1 × 1 0 0 1 × 1 1 , 0 0 0 k g m k g m c m m k g g
  • E 4 , 5 0 0 1 = 4 , 5 0 0 1 × 1 1 0 0 × 1 1 , 0 0 0 k g m k g m m c m k g g

What will she find?

Q4:

Daniel wants to convert 50 m/s into kilometers per hour.

He starts by writing the speed as a ratio of distance to time. What would this look like?

  • A 1 5 0 m s
  • B 1 5 0 s m
  • C 5 0 1 m s
  • D 5 0 2 m s
  • E 5 0 1 s m

Next, he works out how many meters would be covered in 1 hour, which is equal to 3 6 0 0 s. What does he find?

Finally, he converts his result in (2) to kilometers. What speed does he find?

Considering units as algebraic quantities, which of the following is equivalent to Daniel’s conversion?

  • A 5 0 1 = 5 0 1 × 1 3 6 0 0 × 1 0 0 0 1 m s m s s h k m m
  • B 5 0 1 = 5 0 1 × 3 6 0 0 1 × 1 1 0 0 0 m s m s s h k m m
  • C 5 0 1 = 5 0 1 × 6 0 1 × 1 1 0 0 0 m s m s s h k m m
  • D 5 0 1 = 5 0 1 × 3 6 0 0 1 × 1 1 0 0 m s m s s h k m m
  • E 5 0 1 = 5 0 1 × 3 6 0 0 1 × 1 1 0 0 0 m s m s s h k m m

Q5:

Which of the following is correct?

  • A 1 1 = 1 1 × 1 1 0 0 × 1 0 0 0 1 g c m g c m m c m g k g
  • B 1 1 = 1 1 × 1 ( 1 0 ) × 1 0 0 0 1 g c m g c m m c m g k g
  • C 1 1 = 1 1 × 1 0 0 1 × 1 1 0 0 0 g c m g c m c m m k g g
  • D 1 1 = 1 1 × 1 ( 1 0 0 ) × 1 0 0 0 1 g c m g c m m c m g k g
  • E 1 1 = 1 1 × ( 1 0 0 ) 1 × 1 1 0 0 0 g c m g c m c m m k g g

If the density of a material is 𝑑 g/cm3, what is its density expressed in kg/m3?

  • A 1 0 0 0 𝑑 kg/m3
  • B 1 0 𝑑 kg/m3
  • C 𝑑 kg/m3
  • D 𝑑 1 0 kg/m3
  • E 𝑑 1 0 0 0 kg/m3

Q6:

Which of the following is correct?

  • A 1 1 = 1 1 × 1 6 0 × 1 0 0 0 1 m s m s h s m k m
  • B 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 1 m s m s h s m k m
  • C 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 0 m s m s s h k m m
  • D 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 0 1 m s m s h s m k m
  • E 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 m s m s s h k m m

Convert 1 m/s into kilometers per hour, giving your answer as a fraction in its simplest form.

  • A 1 / = 1 8 5 / m s k m h
  • B 1 / = 1 3 6 / m s k m h
  • C 1 / = 5 0 3 / m s k m h
  • D 1 / = 5 1 8 / m s k m h
  • E 1 / = 3 6 / m s k m h

Convert 𝑎 m/s into kilometers per hour.

  • A 1 8 5 𝑎 km/h
  • B 3 6 𝑎 km/h
  • C 5 0 3 𝑎 km/h
  • D 1 3 6 𝑎 km/h
  • E 5 1 8 𝑎 km/h

Q7:

The density 𝐷 of a material can be measured in kilograms per cubic meter, often written as kg/m3. Use dimensional analysis to determine which of the following formulas is correct to calculate the density of a given object, where 𝑚 is its mass and 𝑉 its volume.

  • A 𝐷 = 𝑉 𝑚
  • B 𝐷 = 𝑚 𝑉
  • C 𝐷 = 𝑚 𝑉
  • D 𝐷 = 1 𝑚 𝑉
  • E 𝐷 = 𝑚 ( 𝑉 )

Q8:

When heating up a meal, some parts of the meal can be hot before some others. For instance, chicken will heat up faster than broccoli. This is because different substances have different specific heat capacities, which is the energy needed to raise the temperature of one kilogram of a substance by one degree Celsius. The specific heat capacity is often expressed in joules per kilogram - degree Celsius (or J k g C / ). Using dimensional analysis, find the formula for the amount of energy 𝑄 needed to raise the temperature of a mass 𝑚 of a substance of specific heat capacity 𝑐 by Δ 𝑇 degrees.

  • A 𝑄 = 𝑚 𝑐 Δ 𝑇
  • B 𝑄 = 𝑚 𝑐 Δ 𝑇
  • C 𝑄 = 𝑚 𝑐 Δ 𝑇
  • D 𝑄 = 𝑐 𝑚 Δ 𝑇
  • E 𝑄 = 𝑚 Δ 𝑇 𝑐

Q9:

The amount of pressure an object is exerting is given by 𝑃 = 𝐹 𝐴 , where 𝐹 is the force exerted by the object and 𝐴 is the surface area on which the force is spread. When 𝐹 is measured in newtons, which can be expressed as kg⋅m/s2, and 𝐴 is measured in square meters, the pressure is measured in pascals. Which of the following is the expression of pascals in terms of the base units kilograms, meters, and seconds?

  • A k g m s /
  • B k g s m /
  • C k g m s / ( )
  • D k g s m /
  • E k g s m /

Q10:

Which of the following is correct?

  • A 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 k m h k m h s h m m
  • B 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 0 1 k m h k m h h s m k m
  • C 1 1 = 1 1 × 1 3 6 0 0 × 1 0 0 1 k m h k m h h s m k m
  • D 1 1 = 1 1 × 6 0 1 × 1 1 0 0 0 k m h k m h s h k m m
  • E 1 1 = 1 1 × 3 6 0 0 1 × 1 1 0 0 0 k m h k m h s h m m

Convert 1 km/h into meters per second, giving your answer as a fraction in its simplest form.

  • A 1 / = 1 8 5 / k m h m s
  • B 1 / = 5 1 8 / k m h m s
  • C 1 / = 1 3 6 / k m h m s
  • D 1 / = 3 5 0 / k m h m s
  • E 1 / = 3 6 / k m h m s

Convert 𝑎 km/h into meters per second.

  • A 3 6 𝑎 m/s
  • B 1 3 6 𝑎 m/s
  • C 1 8 5 𝑎 m/s
  • D 5 1 8 𝑎 m/s
  • E 3 5 0 𝑎 m/s

Q11:

Which of the following is correct?

  • A 1 1 = 1 1 × 1 1 0 0 × 1 1 0 0 0 k g m k g m m c m k g g
  • B 1 1 = 1 1 × ( 1 0 0 ) 1 × 1 1 0 0 0 k g m k g m c m m k g g
  • C 1 1 = 1 1 × 1 0 0 1 × 1 1 0 0 0 k g m k g m c m m k g g
  • D 1 1 = 1 1 × 1 ( 1 0 0 ) × 1 0 0 0 1 k g m k g m m c m g k g
  • E 1 1 = 1 1 × 1 1 0 0 × 1 0 0 0 1 k g m k g m m c m g k g

If the density of a material is 𝑑 kg/m3, what is its density expressed in g/cm3?

  • A 1 0 𝑑 g/cm3
  • B 𝑑 1 0 0 0 0 0 g/cm3
  • C 𝑑 1 0 0 0 g/cm3
  • D 𝑑 1 0 g/cm3
  • E 1 0 0 0 𝑑 g/cm3

Q12:

Natalie wants to convert 30 km/h into meters per second

Which of the following is correct?

  • A 3 0 1 = 3 0 1 × 1 6 0 × 1 0 0 0 1 k m h k m h h s m k m
  • B 3 0 1 = 3 0 1 × 3 6 0 0 1 × 1 1 0 0 0 k m h k m h s h k m m
  • C 3 0 1 = 3 0 1 × 3 6 0 0 1 × 1 1 0 0 0 k m h k m h s h k m m
  • D 3 0 1 = 3 0 1 × 6 0 1 × 1 1 0 0 0 k m h k m h s h k m m
  • E 3 0 1 = 3 0 1 × 1 3 6 0 0 × 1 0 0 0 1 k m h k m h h s m k m

What does she find?

Q13:

Einstein’s famous formula states that the equivalent energy ( 𝐸 ) of an object is proportional to its mass 𝑚 as described by 𝐸 = 𝑚 𝑐 , where 𝑐 is the speed of light. When the mass of the object is measured in kilograms and the speed of light in meters per second, the energy is measured in joules. Which of the following is the expression of joules in terms of the base units kilograms, meters, and seconds?

  • A kg⋅m/s
  • B kg⋅m2/s2
  • C kg⋅m/s2
  • D kg⋅m2/s
  • E(kg⋅m/s)2

Q14:

The formula to calculate the density of a material of mass 𝑚 and volume 𝑉 is given by 𝐷 = 𝑚 𝑉 . Which of the following is a possible unit of density?

  • A(m/kg)3
  • B m3/kg
  • C kg/m3
  • D kg/m
  • E(kg/m)3

Q15:

Convert 400 km/h to meters per second, giving your answer to one decimal place.

Q16:

The density of steel is about 8 0 5 0 kg/m3, and the density of lead is about 11.34 g/cm3.

Convert 11.34 g/cm3 to kilograms per cubic meter.

Hence, by how many kilograms per cubic meter is lead denser than steel?

Q17:

The cuboid in the given figure has a mass of 48 kg.

Work out the density of the cuboid in kilograms per cubic centimeter, giving your answer accurate to three significant figures.

Hint : Density = Mass / Volume.

Convert your answer to grams per cubic centimeter.

Q18:

The cylinder in the given figure has a mass of 1 2 4 0 0 kg.

Work out the density of the cylinder in kilograms per cubic meter, giving your answer accurate to three significant figures.

Hint : Density = Mass / Volume.

Convert your answer to grams per cubic centimeter.

Q19:

Which is faster, 28 km/h or 8 m/s?

Q20:

Emma’s friend Daniel told her that his average running speed is 8 mph. Emma knows that she can run at an average of 12 ft/s.

Convert 8 mph to feet per second, giving your answer to two decimal places.

Who is faster? Moreover, by how much?

  • AEmma, by 8.42 ft/s
  • BEmma, by 0.27 ft/s
  • CEmma, by 6.52 ft/s
  • DEmma, by 0.80 ft/s
  • EEmma, by 5.22 ft/s

Q21:

Find the speed of a helicopter flying at 187 miles per hour. Give the answer in miles per second to two decimal places.

Q22:

Daniel usually runs at a speed of 7.7 m/s. Express his speed in feet per second to the nearest hundredth.

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