(for a better understanding of engineering dynamics problems ,under graduate coarse)

1. ABSTRACT:
To analyze the motion of a bicyclist as described in the above problem. I will be making explicit relation
of time with the given velocity-distance relation, and plotting the acceleration, velocity and distance
against time.
ANALYSIS:
S=0 to 40 meters:
 Velocity is dependent on distance as follows:
V=0.25*s
푑푠
푑푡
= 0.25 ∗ 푠, => 1
0.25
∫
푑푠
푠
푡
푡0
= ∫ 푑푡
푠
푠0
t-t0 =
1
0.25
(ln(s)-ln(s0))
s-s0 =0.25*(푒푡 − 푒푡0
) =>> @s0=0, t0
t=4*ln(s)
S=0.25*푒푡 − 0.25
푑푠
푑푡
= 0.25*et =V
푑^2푠
푑푡^2
=0.25*et =a

2.  To get the time for bicyclist to travel 40 meters :
t40=4*(ln(40))
t40=14.755 sec
GRAPH:
15
10
5
0
v-s
0 10 20 30 40 50
velocity
distance
Series1
800000
600000
400000
200000
0
v-t(s)
-5 0 5 10 15 20
-200000
velocity
time
Series1

3. 30
25
20
15
10
5
0
a-s
0 10 20 30 40 50
700000
600000
500000
400000
300000
200000
100000
0
a-t(s)
-5 0 5 10 15 20
Observations:
Here the distance of very infinitesimal range is being covered before the infinitesimal time interval.
 The motion of this bicyclist seems to be recorded as a sudden motion due to which the time
factor is defined at the velocity of 0.25m/s not when the distance was one meter.
 This motion depicts the motion with variable velocity and acceleration at some time t and some
distance s.
 The motion is depicting a very rapid increase of velocity and acceleration which seems
impossible for a human to ride or even a robot cannot run a cycle with this characteristics.
 This motion can be valid for limited time and short distance, as then it could depict the
motion of an Olympian cyclist competing in the event.
acceleration
distance
Series1
-100000
acceleration
time
Series1

4. Conclusion:
This data might be related to some Olympic game researcher or a hypothetical over plotting of the
bicyclist who has a jerky start in the beginning of his ride.
ANNEXURE:
TABLE 1:
s t(s) v(t) a(t) a(s) v(s)
0 #NUM! #NUM! #NUM! 0 0
1 0 0.25 0.25 0.625 0.25
2 2.772589 4 4 1.25 0.5
3 4.394449 20.25 20.25 1.875 0.75
4 5.545177 64 64 2.5 1
5 6.437752 156.25 156.25 3.125 1.25
6 7.167038 324 324 3.75 1.5
7 7.783641 600.25 600.25 4.375 1.75
8 8.317766 1024 1024 5 2
9 8.788898 1640.25 1640.25 5.625 2.25
10 9.21034 2500 2500 6.25 2.5
11 9.591581 3660.25 3660.25 6.875 2.75
12 9.939627 5184 5184 7.5 3
13 10.2598 7140.25 7140.25 8.125 3.25
14 10.55623 9604 9604 8.75 3.5
15 10.8322 12656.25 12656.25 9.375 3.75
16 11.09035 16384 16384 10 4
17 11.33285 20880.25 20880.25 10.625 4.25
18 11.56149 26244 26244 11.25 4.5
19 11.77776 32580.25 32580.25 11.875 4.75
20 11.98293 40000 40000 12.5 5
21 12.17809 48620.25 48620.25 13.125 5.25
22 12.36417 58564 58564 13.75 5.5
23 12.54198 69960.25 69960.25 14.375 5.75
24 12.71222 82944 82944 15 6
25 12.8755 97656.25 97656.25 15.625 6.25
26 13.03239 114244 114244 16.25 6.5
27 13.18335 132860.3 132860.3 16.875 6.75
28 13.32882 153664 153664 17.5 7
29 13.46918 176820.3 176820.3 18.125 7.25
30 13.60479 202500 202500 18.75 7.5
31 13.73595 230880.3 230880.3 19.375 7.75
32 13.86294 262144 262144 20 8
33 13.98603 296480.3 296480.3 20.625 8.25
34 14.10544 334084 334084 21.25 8.5

5. 35 14.22139 375156.3 375156.3 21.875 8.75
36 14.33408 419904 419904 22.5 9
37 14.44367 468540.3 468540.3 23.125 9.25
38 14.55034 521284 521284 23.75 9.5
39 14.65425 578360.3 578360.3 24.375 9.75
40 14.75552 640000 640000 25 10