Съдържание

EuPhO 2024, Експеримент — Задача 5. Task E.5 - Interaction time

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от EuPhO 2024

"EuPhO 2024, Kutaisi, Georgia — Experimental Problems" · 2,5 т.

Условие

With the result from the previous task, it would be possible to determine the interaction time $\tau$ of the more elastic ball with the wooden surface. However, for the less elastic ball, there is no simple description such as Eq. 1. Alternatively, we can make the following assumption.

If, for a certain collision speed $v_0$ , the force experienced by the ball as a function of time can be described as $F_0(t) = f(t)$ , then for any other speed $v_1$ , the time dependence takes a similar shape and we can express the force as:

F_1(t) = a_1 f(b_1 t)\,.

Estimate and plot how the interaction time $\tau$ scales with the collision speed $v$ for collisions of the less elastic rubber ball with a solid surface.

Решение

Task E.5 - Interaction time

When the less elastic ball bounces, we can estimate the change of its momentum $\Delta p_1$ as:

\Delta p_1 = \int_t F_1(t)\mathrm{d}t = \frac{a_1}{b_1}\int f(t)\mathrm{d}t \propto F_{1\max}\tau_1,

where $F_{1\max} = a_1 F_{0\max}$ is the maximal force during the collision and $\tau_1 \approx \tau_0/b_1$ is the interaction time.

In other words,

\frac{\Delta p_1}{\Delta p_0} = \frac{v_1}{v_0} = \frac{F_{1\max}}{F_{0\max}}\cdot\frac{\tau_1}{\tau_0}.

By measuring $v_1/v_0 = \sqrt{h_1/h_0}$ and $\frac{F_{1\max}}{F_{0\max}} = \frac{U_1}{U_0}$ , we can estimate the time scaling $\tau_1/\tau_0$ :

\frac{U_1}{U_0} = \frac{\tau_1}{\tau_0}\cdot\frac{\sqrt{h_1}}{\sqrt{h_0}}, \qquad \text{or} \qquad \tau \propto \frac{U}{\sqrt{h}}

The measurements are in the table 7. We plot the dependency of $U/\sqrt{h_1}$ on $\sqrt{h_1}$ and find that it is horizontal up to the measurement errors (Fig. 12). In other words, $\tau$ does not change too much within the range of forces.

Plot of δ · v_collision / a.u. (0.00 to 0.05) versus γ · τ / a.u. (about 3 to 18): experimental values with error bars joined by a line with a shaded band, roughly horizontal around 0.03; legend 'experimental values'. — Figure 12

E.5 - Interaction time	Points
Theory	0.5
find how $a_1/b_1$ depends on $v$	0.3
express it through measurable ratios	0.2
Measurements and plots	1.5
0.1 per measured force value, up to 7 different force values	0.7
0.1 per repetition per data point, up to 3 repetitions	0.3
Plot	0.2
coordinates are related to theory	0.3
Conclusion ( $\tau$ doesn't depend on $v$ )	0.5
Total on E.5	2.5

Raw data tables

Table 1: Data for 1

$S$	364	305	250	200
$s1$	286	237	195	152
$s2$	286	241	193	151
$s3$	287	239	194	154
$s4$	282	238	196	155
$s5$	286	237	193	155
$h$	373	314	259	209
$h1$	290	241	199	156
$h2$	290	245	197	155
$h3$	291	243	198	158
$h4$	286	242	200	159
$h5$	290	241	197	159

Table 2: Data for 2a

$U$	$t$
mV	s
1408	0
1138	5
943	10
762	15
623	20
502	25
424	30
333	35
276	40
223	45
180	50
151	55
124	60
102	65
81	70

Table 3: Data for 2b

charges	$U_c$
	mV
1	15
2	31
3	43
4	64
5	78
6	86
7	97
8	117
9	138
10	142
15	206
20	261
25	322
30	372

Table 4: Data for 2c

$m$	$U1$	$U2$	$U3$	$U4$	$U5$
kg	mV	mV	mV	mV	mV
0.0	0	0	0	0	0
0.1	30	34	39	42	37
0.2	63	78	87	82	77
0.3	126	110	119	119	116
0.4	142	173	153	156	153
0.5	176	206	187	186	206
0.6	240	224	216	249	230
0.7	286	285	259	270	272
0.8	304	295	306	306	296
0.9	335	328	337	338	334
1.0	367	346	373	357	339
1.3	429	390	447	432	458
1.7	557	576	550	541	575
2.0	620	629	625	635	600
2.3	702	711	711	676	671
2.7	812	798	759	762	787
3.0	823	847	820	838	841
3.3	856	855	840	888	854
3.7	994	911	926	916	884
4.0	977	881	881	887	911
4.5	1060	1024	830	799	812
5.0	977	918	852	893	766
5.5	954	817	789	820	839
6.0	802	890	907	1089	942
6.5	892	736	836	903	842
7.0	985	867	859	815	795
7.5	984	968	833	1068	981
8.0	1110	993	1247	1022	1061
8.5	988	814	971	1003	874
9.0	1181	1267	944	1022	907

Table 5: Data for 3

$m$	$U1$	$U2$	$U3$	$U4$	$U5$	$U6$
kg	mV	mV	mV	mV	mV	mV
0.15	52	50	49	55	52	56
0.3	105	102	98	109	111	110
0.5	175	178	166	184	170	172
0.7	245	250	258	233	242	239
1.0	355	340	354	351	345	357

Table 6: Data for 4

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
363	334	437	411	410	424	422
341	312	399	395	380	398	413
320	291	411	414	414	378	425
300	271	337	358	423	385	369
280	251	356	345	393	401	358
260	231	320	311	361	461	373
240	211	303	364	361	332	371
220	191	315	340	329	287	365
200	171	332	326	336	345	347
180	151	305	322	280	316	315
160	131	297	297	284	295	273
140	111	276	271	269	260	261
120	91	253	238	235	241	252
100	71	215	203	205	217	217
80	51	178	177	176	180	181
60	31	150	141	138	146	146
40	11	103	103	116	102	104

Table 7: Data for 5

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
40	11	106	103	103	104	103
60	31	162	175	153	172	153
100	71	240	252	227	249	284
140	111	320	312	367	328	330
160	131	399	387	361	388	403
200	171	456	450	451	457	411
220	191	464	485	484	486	448
240	211	482	494	514	497	498
280	251	554	555	567	549	550
320	291	606	602	507	569	550
360	331	660	628	645	652	650

Оригинал в Архива: eupho2024_experiment_problems.pdf · официални решения: eupho2024_experiment_solutions.pdf

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
363	334	437	411	410	424	422
341	312	399	395	380	398	413
320	291	411	414	414	378	425
300	271	337	358	423	385	369
280	251	356	345	393	401	358
260	231	320	311	361	461	373
240	211	303	364	361	332	371
220	191	315	340	329	287	365
200	171	332	326	336	345	347
180	151	305	322	280	316	315
160	131	297	297	284	295	273
140	111	276	271	269	260	261
120	91	253	238	235	241	252
100	71	215	203	205	217	217
80	51	178	177	176	180	181
60	31	150	141	138	146	146
40	11	103	103	116	102	104

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
40	11	106	103	103	104	103
60	31	162	175	153	172	153
100	71	240	252	227	249	284
140	111	320	312	367	328	330
160	131	399	387	361	388	403
200	171	456	450	451	457	411
220	191	464	485	484	486	448
240	211	482	494	514	497	498
280	251	554	555	567	549	550
320	291	606	602	507	569	550
360	331	660	628	645	652	650

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
363	334	437	411	410	424	422
341	312	399	395	380	398	413
320	291	411	414	414	378	425
300	271	337	358	423	385	369
280	251	356	345	393	401	358
260	231	320	311	361	461	373
240	211	303	364	361	332	371
220	191	315	340	329	287	365
200	171	332	326	336	345	347
180	151	305	322	280	316	315
160	131	297	297	284	295	273
140	111	276	271	269	260	261
120	91	253	238	235	241	252
100	71	215	203	205	217	217
80	51	178	177	176	180	181
60	31	150	141	138	146	146
40	11	103	103	116	102	104

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
40	11	106	103	103	104	103
60	31	162	175	153	172	153
100	71	240	252	227	249	284
140	111	320	312	367	328	330
160	131	399	387	361	388	403
200	171	456	450	451	457	411
220	191	464	485	484	486	448
240	211	482	494	514	497	498
280	251	554	555	567	549	550
320	291	606	602	507	569	550
360	331	660	628	645	652	650

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
363	334	437	411	410	424	422
341	312	399	395	380	398	413
320	291	411	414	414	378	425
300	271	337	358	423	385	369
280	251	356	345	393	401	358
260	231	320	311	361	461	373
240	211	303	364	361	332	371
220	191	315	340	329	287	365
200	171	332	326	336	345	347
180	151	305	322	280	316	315
160	131	297	297	284	295	273
140	111	276	271	269	260	261
120	91	253	238	235	241	252
100	71	215	203	205	217	217
80	51	178	177	176	180	181
60	31	150	141	138	146	146
40	11	103	103	116	102	104

$S$	$h$	$U1$	$U2$	$U3$	$U4$	$U5$
mm	mm	mV	mV	mV	mV	mV
40	11	106	103	103	104	103
60	31	162	175	153	172	153
100	71	240	252	227	249	284
140	111	320	312	367	328	330
160	131	399	387	361	388	403
200	171	456	450	451	457	411
220	191	464	485	484	486	448
240	211	482	494	514	497	498
280	251	554	555	567	549	550
320	291	606	602	507	569	550
360	331	660	628	645	652	650