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.
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
To calculate the magnitude of the average acceleration, we need to determine the change in velocity and the change in time.
Given:
Velocity vector at time t_1, v_1 = 5.40 m/s (along the y-axis)
Velocity vector at time t_2, v_2 = ?
Change in time, change in t = 0.7 s
To determine v_2, we can use the fact that the poodle is running counter-clockwise in a circle. Since the poodle is on the x-axis at time t_1, the velocity vector v_1 points along the y-axis. This means that at time t_2, the poodle would have moved exactly one-quarter of the circle. Therefore, the velocity vector v_2 would be pointing in the negative x-direction.
Now, we need to find the magnitude of v_2. We know that the poodle is running at a constant speed of 5.40 m/s, and the radius of the circle is 2.3 m. Using the formula for the circumference of a circle (C = 2πr), we can find the distance traveled in one-quarter of the circle:
Distance = (1/4) * circumference = (1/4) * 2π * 2.3 m = 1.15π m
Since the poodle is running at a constant speed, the magnitude of the velocity vector v_2 would be equal to the distance traveled divided by the change in time:
Magnitude of v_2 = Distance / change in t = (1.15π m) / (0.7 s) ≈ 2.346 m/s (to three significant figures)
Finally, we can calculate the magnitude of the average acceleration using the formula:
Average acceleration = (Magnitude of v_2 - Magnitude of v_1) / change in t
Average acceleration ≈ (2.346 m/s - 5.40 m/s) / (0.7 s) ≈ (-3.054 m/s) / (0.7 s) ≈ -4.363 m/s^2
Therefore, the magnitude of the average acceleration, to four significant figures, is approximately 4.363 m/s^2.