University Physics I - Classical Mechanics, 2019

2.2. ACCELERATION 41
2.2.3 Acceleration as a vector
In two (or more) dimensions we introduce the average acceleration vector
⃗a av = Δ⃗v
Δt = 1 Δt (⃗v f − ⃗v i ) (2.11)
whose components are a av,x =Δv x /Δt, etc.. The instantaneous acceleration is then the vector given
by the limit of Eq. (2.11)asΔt → 0, and its components are, therefore, a x = dv x /dt, a y = dv y /dt,...
Note that, since ⃗v i and ⃗v f in Eq. (2.11) are vectors, and have to be subtracted as such, the acceleration
vector will be nonzero whenever ⃗v i and ⃗v f are different, even if, for instance, their magnitudes
(which are equal to the object’s speed) are the same. In other words, you have accelerated motion
whenever the direction of motion changes, even if the speed does not.
As long as we are working in one dimension, I will follow the same convention for the acceleration
as the one I introduced for the velocity in Chapter 1: namely, I will use the symbol a, without
a subscript, to refer to the relevant component of the acceleration (a x ,a y ,...), and not to the
magnitude of the vector ⃗a.
2.2.4 Acceleration in different reference frames
In Chapter 1 you saw that the following relation (Eq. (1.19)) holds between the velocities of a
particle P measured in two different reference frames, A and B:
⃗v AP = ⃗v AB + ⃗v BP (2.12)
What about the acceleration? An equation like (2.12) will hold for the initial and final velocities,
and subtracting them we will get
Δ⃗v AP =Δ⃗v AB +Δ⃗v BP (2.13)
Now suppose that reference frame B moves with constant velocity relative to frame A. In that case,
⃗v AB,f = ⃗v AB,i ,soΔ⃗v AB = 0, and then, dividing Eq. (2.13) byΔt, and taking the limit Δt → 0, we
get
⃗a AP = ⃗a BP (for constant ⃗v AB ) (2.14)
So, if two reference frames are moving at constant velocity relative to each other, observers in both
frames measure the same acceleration for any object they might both be tracking.
The result (2.14) means, in particular, that if we have an inertial frame then any frame moving
at constant velocity relative to it will be inertial too, since the respective observers’ measurements