9702_s22_qp_13 Question 3

The drag coefficient C_d is a number with no units. It is used to compare the drag on different cars at different speeds. C_d is given by the equation

 Cd = 2F / vⁿρA 

where F is the drag force on the car, ρ is the density of the air, A is the cross-sectional area of the car and v is the speed of the car. 

What is the value of n? 

A 1 

B 2 

C 3 

D 4

ANSWER: B

The Physics behind the answer:

• The key idea here is to do dimensional analysis i.e. see what exponent n must be so that the units of 2F is the same as the unit of vⁿρA.
• For the 2F,
• the unit is N
• which is defined as kg m s^-2
• For the vⁿρA,
• the unit of v is (m s^-1)ⁿ, ρ is kg m^-3, and A is m²
• simplifying the above gives (m s^-1)ⁿ kg m^-1
• Combining the above 2 sets of units, we will have 
• the kgs cancel each other
• leading to m² s^-2 / (m s^-1)ⁿ 
• Now, it is clear that n = 2 for these 2 sets of units cancel each other i.e. to be unitless.