Question

For no apparent reason, a poodle is running counter-clockwise at a constant speed of 5.40 m/s in a circle with radius 2.3 m. Let v_1 be the velocity vector at time t_1, and let v_2 be the velocity vector at time t_2. Consider change in v = v_2- v_1 and change in t = t_2 - t_1. Recall that average acceleration = change in v/ change in t.

Hint: It may be helpful to assume that at time t_1, the poodle is on the x-axis, i.e., that the velocity vector \bar{v}_1 points along the y-axis.

For change in t = 0.7 s calculate the magnitude (to four significant figures) of the average acceleration a.

Answer 1

geometry if a = 0 at t = 0
angle from x axis (call it a) = v t/r

Vx = -v sin a
Vy = v cos a

at start a = 0 so
V1x = 0
V1y = v
at time t
V2x = -v sin(vt/r)
V2y = v cos(vt/r)

call change d

d Vx = -v sin(vt/r)
d Vy = v[cos(vt/r)-1]

ax = -(v/t) sin(vt/r)
ay = (v/t)[cos(vt/r)-1]

so for our numbers
v/t = 5.4/.7 = 7.714
vt/r = 1.643 about 94 degrees, quadrant2
ax = -7.714 sin(1.643) = -7.694
ay= 7.714[cos(1.643)-1] = -8.270
|A| =SQRT(ax^2+ay^2) = 11.30 m/s^2

compare to v^2/r
v^2/r = 5.4^2/2.3 = 12.7 not too far off centripetal acceleration