The following table lists the components of a vehicle’s velocity vector and…

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

The following table lists the components of a vehicle’s velocity vector

\mathbf v(t)

and acceleration vector

\mathbf a(t)

at four equally spaced times.

Time $t$ (s)	$v_x$ (m/s)	$v_y$ (m/s)	$a_x$ (m/s $^2$ )	$a_y$ (m/s $^2$ )
0	12	5	$-3$	4
1	10	9	$-2$	3
2	6	12	$-1$	1
3	2	14	0	$-1$

Assume both vectors vary smoothly between the listed times. (a) For each time listed, evaluate

\mathbf v(t)\cdot\mathbf a(t)

. (b) Identify the intervals on which the vehicle’s speed is increasing, decreasing, or momentarily constant. Justify each interval using your results from part (a). (c) Between

t=1

and

t=2

, estimate the smallest angle (in degrees) attained between

\mathbf v(t)

and

\mathbf a(t)

by linearly interpolating the vectors. Show all steps in your estimation procedure, including how the angle is calculated from the interpolated vectors.

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