Converting Seconds to MPH

Date: 07/11/2000 at 19:54:16
From: Bob
Subject: Seconds to MPH conversion CHART

I'm looking for a chart that converts seconds traveled to miles per 
hour.  I saw something in an FAQ about dividing by 3600. Thanks.

Date: 07/12/2000 at 12:07:54
From: Doctor Rick
Subject: Re: Seconds to MPH conversion CHART

Hi, Bob.

Could you specify the problem more carefully? You can't compute a
speed (miles per hour) from a time (seconds) alone. Do you want, 
perhaps, times in seconds across the top and distances in feet down 
the side? Then the (average) speed is the distance divided by the 
time, and we need to convert feet to miles and seconds to hours. The 
division by 3600 converts seconds to hours; we similarly need to 
divide the distance in feet by 5280 to get the distance in miles. The 
whole formula for the problem as I stated it is

     D = distance in feet
     T = time in seconds
     S = speed in miles per hour

     S = (D/5280)/(T/3600)
       = (D/T)*(3600/5280)
       = (15/22)D/T

You can use this formula in a spreadsheet to generate a table with the 
ranges of times and distances of interest to you. For instance, in 
Excel, a formula like

     =15/22*$B3/C$2

can be used to generate a table like

    1      2      3      4      5      6      7
-------------------------------------------------------------
1   0.682  0.341  0.227  0.170  0.136  0.114  0.097
2   1.364  0.682  0.455  0.341  0.273  0.227  0.195
3   2.045  1.023  0.682  0.511  0.409  0.341  0.292
4   2.727  1.364  0.909  0.682  0.545  0.455  0.390
5   3.409  1.705  1.136  0.852  0.682  0.568  0.487
6   4.091  2.045  1.364  1.023  0.818  0.682  0.584
7   4.773  2.386  1.591  1.193  0.955  0.795  0.682
8   5.455  2.727  1.818  1.364  1.091  0.909  0.779
9   6.136  3.068  2.045  1.534  1.227  1.023  0.877
10  6.818  3.409  2.273  1.705  1.364  1.136  0.974
11  7.500  3.750  2.500  1.875  1.500  1.250  1.071
12  8.182  4.091  2.727  2.045  1.636  1.364  1.169
13  8.864  4.432  2.955  2.216  1.773  1.477  1.266
14  9.545  4.773  3.182  2.386  1.909  1.591  1.364
15  10.227 5.114  3.409  2.557  2.045  1.705  1.461
16  10.909 5.455  3.636  2.727  2.182  1.818  1.558

where the numbers across the top are seconds and the numbers at the 
left are feet. Going 11 feet in 3 seconds comes out to 2.5 miles per 
hour.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/12/2000 at 12:16:22
From: Doctor Twe
Subject: Re: seconds to MPH conversion CHART

Hi Bob! Thanks for writing to Dr. Math!

Unfortunately, seconds measures time and mph measures speed, so unless 
we have more information (a fixed distance traveled, for example) we 
can't convert seconds to mph. To convert seconds to hours, we divide 
by 3600 - that's where that number comes in.

Let's say, for example, that we were measuring sprinters' times in the 
100-yard dash and wanted to convert them to mph. We now have a fixed 
distance (100 yards), so we can calculate the speed. Let's say the 
sprinters ran it in 4.4 s, 4.6 s, 4.85 s and 5.2 s.

     s = d / t

where: s = speed
       d = distance
       t = time

So for the first sprinter, we have:

     s1 = 100 yd / 4.4 s
        = 22.73 yd/s

Yards per second? Let's convert that to miles per hour. We can 
convert yards to miles by dividing by 1760 and, as mentioned above, we 
can convert seconds to hours by dividing by 3600. So we have:

     s1 = (100 yd / 1760 yd/mi) / (4.4 s / 3600 s/hr)
        = 0.0568 mi / 0.00122 hr
        = 46.49 mph

Converting each measurement to miles and hours takes extra time. If we
have more of the same calculations to do, we can incorporate the 
constants 1760 and 3600 (and even the 100 yds, if it doesn't change) 
directly into the formula with a little algebra:

              100 yd
            ----------
            1760 yd/mi
     s = ----------------
                t
            ---------
            3600 s/hr

           100 yd     3600 s/hr
       = ---------- * ---------
         1760 yd/mi       t

         360000 mi
       = ---------
         1760*t hr

     s = 204.54/t mph   [or s = 2250/(11*t) mph for more accuracy.]

Using this "condensed" formula, we can quickly convert the other 
sprinters' times into speeds:

     s2 = 204.54 / 4.6
        = 44.47 mph

     s3 = 204.54 / 4.85
        = 42.17 mph

     s4 = 204.54 / 5.2
        = 39.34 mph

I hope this helps! If you have any more questions, write back!

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/12/2000 at 15:33:50
From: Bob Hubert
Subject: Re: seconds to MPH conversion CHART

Thanks for the answer you provided. I think my question has 
complicated the issue. Let me be more specific and you tell me which 
formula is best. I thought it would be very simple to locate a chart 
in which the math has already been computed.

If you travel 1 mile in 60 seconds, you are traveling 60 miles per 
hour. If you travel that mile in 55 seconds, what is your speed in 
MPH?

I want a chart that people can carry in their cars so they do not 
depend upon their unreliable automobile speedometers. They can use 
a watch to time how many seconds it takes to travel from one milepost 
to another, then look at this chart to correctly determine their 
speed. I'm playing with the numbers and 3600 keeps coming into the 
equation. I'm so close but it is driving me a little nuts that I can't 
solve this.

Thanks.

Date: 07/12/2000 at 16:19:12
From: Doctor Rick
Subject: Re: seconds to MPH conversion CHART

Hi, Bob. Thanks for clarifying your question. If you had said "I'm 
looking for a chart that converts seconds IT TAKES TO TRAVEL ONE MILE 
to miles per hour," both Dr. TWE and I could have saved our effort.

The formula is almost there in my response. Speed is distance divided 
by time. If the distance is 1 mile and the time is T seconds, then the 
time in hours is T/3600, and the speed is

     Speed = (1 mile)/(T/3600 hours)
           = 3600/T

That's pretty simple; you just had it upside down. The table looks 
like this:

Seconds  MPH
------------
36       100.0
37       97.3
38       94.7
39       92.3
40       90.0
41       87.8
42       85.7
43       83.7
44       81.8
45       80.0
46       78.3
47       76.6
48       75.0
49       73.5
50       72.0
51       70.6
52       69.2
53       67.9
54       66.7
55       65.5
56       64.3
57       63.2
58       62.1
59       61.0
60       60.0
61       59.0
62       58.1
63       57.1
64       56.3
65       55.4
66       54.5
67       53.7
68       52.9
69       52.2
70       51.4
71       50.7
72       50.0
73       49.3
74       48.6
75       48.0
76       47.4
77       46.8
78       46.2
79       45.6
80       45.0
81       44.4
82       43.9
83       43.4
84       42.9
85       42.4
86       41.9
87       41.4
88       40.9
89       40.4
90       40.0
91       39.6
92       39.1
93       38.7
94       38.3
95       37.9
96       37.5
97       37.1
98       36.7
99       36.4
100      36.0
101      35.6
102      35.3
103      35.0
104      34.6
105      34.3
106      34.0
107      33.6
108      33.3
109      33.0
110      32.7
111      32.4
112      32.1
113      31.9
114      31.6
115      31.3
116      31.0
117      30.8
118      30.5
119      30.3
120      30.0
121      29.8
122      29.5
123      29.3
124      29.0
125      28.8
126      28.6
127      28.3
128      28.1
129      27.9
130      27.7
131      27.5
132      27.3
133      27.1
134      26.9
135      26.7
136      26.5
137      26.3
138      26.1
139      25.9
140      25.7
141      25.5
142      25.4
143      25.2
144      25.0

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/12/2000 at 17:44:55
From: Doctor Twe
Subject: Re: seconds to MPH conversion CHART

Hi again Bob! Thanks for writing back!

As with the sprinters example, here we have a fixed distance. In this 
case, it's even easier than the 100 yards - it's exactly one mile! We 
can use the formula I gave previously to construct the chart:

     s = d / t

Where: s = speed
       d = distance (1 mile)
       t = time (in seconds)

Since the time is in seconds, we can convert it to hours by dividing 
it by 3600. But dividing the denominator (bottom of the fraction) by 
3600 is the same as multiplying the numerator (top of the fraction) by 
3600, so we get:

           1 mi
     s = --------
         (t/3600)

         1 mi * 3600
       = -----------
              t

     s = 3600/t

So to convert the time (in seconds) it takes to travel 1 mile to speed 
in mph, just divide 3600 by the number of seconds. As a chart, we 
have:

     Time   Speed          Time   Speed          Time   Speed
     (s)    (mph)          (s)    (mph)          (s)    (mph)
     ----   -----          ----   -----          ----   -----
      30     120            60      60            90      40
      31     116            61      59            91      40
      32     113            62      58            92      39
      33     109            63      57            93      39
      34     106            64      56            94      38
      35     103            65      55            95      38
      36     100            66      55            96      38
      37      97            67      54            97      37
      38      95            68      53            98      37
      39      92            69      52            99      36
      40      90            70      51           100      36
      41      88            71      51           101      36
      42      86            72      50           102      35
      43      84            73      49           103      35
      44      82            74      49           104      35
      45      80            75      48           105      34
      46      78            76      47           106      34
      47      77            77      47           107      34
      48      75            78      46           108      33
      49      73            79      46           109      33
      50      72            80      45           110      33
      51      71            81      44           111      32
      52      69            82      44           112      32
      53      68            83      43           113      32
      54      67            84      43           114      32
      55      65            85      42           115      31
      56      64            86      42           116      31
      57      63            87      41           117      31
      58      62            88      41           118      31
      59      61            89      40           119      30

Using the formula above, you can extend this chart as necessary. Note 
that at higher speeds, the change in mph per second increases, so the 
margin of error also increases.

Personally, I wouldn't put much stock in the reliability of this 
method. It relies on the precision of starting and stopping the watch 
at the precise moment you pass the mile marker (parallax becomes an 
issue), and the accuracy of the distance between the mileposts. Also, 
I'm not so sure it's a good idea to try to read your watch while 
driving - perhaps a passenger could do this.

I hope this helps. If you have any more questions, write back again.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/