A race car moves such that its position fits the relationship x = (4.0 m/s) t +
ID: 1881221 • Letter: A
Question
A race car moves such that its position fits the relationship
x = (4.0 m/s)t + (0.80 m/s3)t3
where x is measured in meters and t in seconds.
(a) A plot of the car's position versus time is which of the following?
(b) Determine the instantaneous velocity of the car at
t = 3.6 s,
using time intervals of 0.40 s, 0.20 s, and 0.10 s. (In order to better see the limiting process keep at least three decimal places in your answer.)
t = 0.40 s
(Use the interval from t = 3.40 s to 3.80 s.)
t = 0.20 s
(Use the interval from t = 3.50 s to 3.70 s.)
t = 0.10 s
(Use the interval from t = 3.55 s to 3.65 s)
(c) Compare the average velocity during the first 3.6 s with the results of part (b).
The average velocity of m/s is _____________ (much less than, about the same, as much, or greater than) the instantaneous velocity
Please show work
x (m) 240 220- 200 180 160 140 120 100 80 60 40 20 - t (s) 1.00 2.00 3.00 4.00 5.00 6.00Explanation / Answer
a)
Substituting t = 6 s,
x = (4.0 m/s) x 6 + (0.80 m/s3) x (6)3
= 24 + 172.8
= 196.8 m
So third graph (first column, second row) is the correct one.
b)
Instantaneous velocity, v = dx/dt
= 4 + 0.8 x 3t2
= 4 + 2.4 t2
At t = 3.6s,
v = 4 + 2.4 x (3.6)2
= 35.104 m/s.
i) t = 0.4 s
vi at 3.4 s = 4 + 2.4 x (3.4)2
= 31.744 m/s.
vf at 3.8 s = 4 + 2.4 x (3.8)2
= 38.656 m/s.
Average velocity = 1/2 [vi + vf]
= 0.5 x [31.744 + 38.656]
= 35.200 m/s
ii) t = 0.2 s
vi at 3.5 s = 4 + 2.4 x (3.5)2
= 33.4 m/s.
vf at 3.7 s = 4 + 2.4 x (3.7)2
= 36.856 m/s.
Average velocity = 1/2 [vi + vf]
= 0.5 x [33.4 + 36.856]
= 35.128 m/s
iii) t = 0.1 s
vi at 3.55 s = 4 + 2.4 x (3.55)2
= 34.246 m/s.
vf at 3.65 s = 4 + 2.4 x (3.65)2
= 35.974 m/s.
Average velocity = 1/2 [vi + vf]
= 0.5 x [34.246 + 35.974]
= 35.110 m/s
So t = 0.1 s is the best limiting condition.
c)
Total distance travelled in first 3.6 s is,
x = (4.0 m/s) x 3.6 + (0.80 m/s3) x (3.6)3
= 51.7248 m
Average velocity = 51.7248/3.6
= 14.368 m/s
Average velocity is much less than the instantaneous velocity.