The previous demonstration has been made under the assumption, many times repeated, that the frictional resistances to the movement of the train are zero. The same law will hold if we may assume that the engine is doing an amount of work which at all times is just equal to that required to overcome such resistances. It has been found that the tractive resistances (which here include all resistances except those due to grade) are nearly independent of velocity for a very considerable range of velocity, which includes the most common freight-train velocities. It is also assumed that the draw-bar pull is uniform for these various velocities. This last assumption is virtually the same as assuming that the tractive power of the drivers is independent of velocity, and that the engine is capable of varying its output measured in horse-power indefinitely. None of these assumptions are strictly true, but a thorough appreciation of this method of calculation will assist very materially in studying the value and use of momentum grades, since the error is practically inappreciable when operating small sags and humps, and does not become of very great value except in extreme cases. We will first apply the method to some practical cases on the basis, as before stated, that the tractive resistances are independent of velocity, and that the pull on the draw-bar of the locomotive is constant. Assume that a train is passing A (see Fig. 24) running at a velocity of 15 miles per hour. Assume that the throttle is not changed nor any brakes applied, and that the engine is capable of increasing its horse-power, so that, in spite of its increased velocity on the succeeding down grade, it is still able to exert the same draw-bar pull. At A its velocity head is that due to 15 miles per hour or 7.90 feet. At B it has gained 20 feet more, and its velocity is that due to a velocity head of 27.90 feet, or nearly 28.2 miles per hour. Upon climbing the grade BC, when it reaches the point B', it has given up its velocity head, due to the additional 20 feet, and its velocity head is again 7.90. At the point C, which is 4 feet higher than B', its velocity head is only 3.90, which corresponds to a speed of about 10.5 miles per hour. As the train starts down the grade CD its velocity continues to increase from 10.5 miles per hour, and when it has reached C' it has again recovered the 4 feet of velocity head and will again be moving at the velocity of 15 miles per hour. If at this point the grade again becomes level, the train will continue to move on as before at a velocity of 15 miles per hour. It will have been practically uninfluenced by the presence of the combined sag and hump.

Table XX. Velocity Head (Representing The Kinetic Energy) Of Trains Moving At Various Velocities

Velocity m. per h.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

10

3.51

3.58

3.65

3.75

3.79

) 3.87

' 3.95

4.02

4.1C

4.17

11

4.25

4.33

4.41

4.49

) 4.57

4.65

4.73

4.81

4.89

4.97

12

5.06

5.15

5.22

5.32

5.41

5.50

5.58

5.67

5.75

5.84

13

5.93

6.02

6.12

6.21

. 6.31

6.40

6.50

6.59

6.69

6.78

14

6.88

6.98

7.08

7.19

7.29

7.39

7.49

7.60

7.70

7.80

15

7.90

8.00

8.11

8.22

8.33

8.44

8.55

8.66

8.77

8.88

16

8.99

9.10

9.21

9.32

! 9.43

9.55

9.67

9.79

9.91

10.03

17

10.15

10.27

10.39

10.51

10.63

10.75

10.87

10.99

11.12

11.25

18

11.38

11.50

11.63

11.76

11.89

12.02

12.15

12.28

12.41

12.55

19

12.68

12.81

12.95

13.08

13.22

13.35

13.49

13.63

13.77

13.91

20

14.05

14.19

14.33

14.47

14.61

14.75

14.89

15.04

15.19

15.34

21

15.49

15.64

15.79

15.94

16.09

16.24

16.39

[16.54

16.69

16.84

22

17.00

17.15

17.30

17.46

17.62

17.78

17.94

18.10

18.26

18.42

23

18.58

18.74

18.90

19.06

19.22

19.38

19.55

19.72

19.89

20.06

24

20.23

20.40

20.57

20.74

20.91

21.08

21.25

21.42

21.59

21.77

25

21.95

22.12

22.30

22.48

22.66

22.84

23.02

23.20

23.38

23.56

26

23.74

23.92

24.10

24.28

24.46

24.65

24.84

25.03

25.22

25.41

27

25.60

25.79

25.98

26.17

26.36

26.55

26.74

26.93

27.13

27.33

28

27.53

27.73

27.93

28.13

28.33

28.53

28.73

28.93

29.13

29.33

29

29.53

29.73

29.93

30.13

30.34

30.55

30.76

30.97

31.18

31.39

30

31.60

31.81

32.02

32.23

32.44

32.65

32.86

33.08

33.30

33.52

31

33.74

33.96

34.18

34.40

34.62

34.84

35.06

35.28

35.50

35.72

32

35.95

36.17

36.39

36.62

36.85

37.08

37.31

37.54

37.77

38.00

33

38.23

38.46

38.69

38.92

39.15

39.38

39.62

39.86

40.10

40.34

34

40.58

40.82

41.06

41.30

41.54

41.78

42.02

42.26

42.51

42.76

35

43.01

43.26

43.51

43.76

44.01

44.26

44.51

44.76

45.01

45.26

36

45.51

45.76

46.01

46.26

46.52

46.78

47.04

47.30

47.56

47.82

37

48.08

48.34

48.60

48.86

49.12

49.38

49.64

49.91

50.18

50.45

38

50.72

50.99

51.26

51.53

51.80

52.07

52.34

52.61

52.88

53.15

39

53.42

53.69

53.96

54.23

54.51

54.79

55.07

55.35

55.63

55.91

40

56.19

56.47

56.75

57.03

57.31

57.59

57.87

58.16

58.45

58.74

41

59.03

59.32

59.61

59.90

60.19

60.48

60.77

61.06

61.35

61.64

42

61.94

62.23

62.52

62.82

63.12

63.42

63.72

64.02

64.32

64.62

43

64.92

65.22

65.52

65.82

66.12

66.43

66.74

67.05

67.36

67.67

44

67.98

68.29

68.60

68.91

69.22

69.53

69.84

70.15

70.46

70.78

45

71.10

71.42

71.74

72.06

72.38

72.70

73.02

73.34

73.66

73.98

46

74.30

74.62

74.94

75.26

75.59

75.92

76.25

76.58

76.91

77.24

47

77.57

77.90

78.23

78.56

78.89

79.22

79.55

79.89

80.23

80.57

48

80.91

31.25

81.59

81.93

82.27

82.61

82.95

83.29

83.63

83.97

49

34.32

34.66

85.00

85.34

85.69

86.04

86.39

86.74

87.09

87.44

50

87.79

38.14

88.49

88.85

89.20

89.55

89.91

90.26

90.61

90.97

Relation of virtual and actual profile through a sag and over a hump.

Fig. 24. Relation of virtual and actual profile through a sag and over a hump.