Acceleration Problem Examples
Here’s a couple typical problems involving the decomposition of acceleration into tangen-
tial and normal components.
1. Problem: (From class handout on Wednesday 4/1.) Suppose a particle moves in 2D
space with position r(t) =< R cos(qt/R), R sin(qt/R)>at time t, where qand Rare
some positive constants. Compute each of v(t),kv(t)k,T(t),a(t),N(t), and κ(t) (as
functions of t, though some may be constant). Write out aT, aN,aT,and aN. Then
compute aTand aNdirectly from the vector projection formulas and compare.
Solution: It’s easy to compute directly that
v=<−sin(qt/R),cos(qt/R)>
kvk= 1
a=<−cos(qt/R)/R, −sin(qt/R)/R >
T=<−sin(qt/R),cos(qt/R)>(same as v)
N=<−cos(qt/R)/R, −sin(qt/R)/R >
κ= 1/R.
The formula a=d(kvk)
dt T+κkvk2Nthen yields
a= 0T+1
RN=<−cos(qt/R)/R, −sin(qt/R)/R >
which is of course the same as above. Note that aT=d(kvk)
dt = 0 since the speed is
constant—there is no tangential acceleration here, and so aT=0(the zero vector).
Also, aN= 1/R; all of the acceleration is normal, due to changing direction, so that
a=aN.
You can also compute directly that
aT=a·v
v·vv=0
1v=0
aN=a−aT=a=<−cos(qt/R)/R, −sin(qt/R)/R > .
2. Another Problem: A particle moves along a path parameterized by the 3D curve
x=u−1, y =u2+ 2, z =−u, moving in the direction of increasing x; note that
here uis just some parameter, not necessarily time. When the particle is at the point
(0,3,0) (corresponding to u= 1) it has a speed of 2 meters per second, and its speed
is increasing at a rate of 5 meters per second squared. Find the acceleration of the
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