29a. Constraints from m_cloud and v_cloud —Θ_cl = 30°

    The results are presented for weights of 5, 6, 7, and 8 lb in order, with a summary at the end. All the allowable answers give negative lurches, and all the ranges include the observed -0.8 ft s^-1. The lurches for the 8-lb head are less negative than for the other weights, however.
    Each subsection is set up with the results for PE first and those for v_lurch next. The allowed PEs are shown in bold green, the disallowed values in red. That masking is then applied to the results for v_lurch.

For m_head = 5 lb
    The allowed PEs are 110–600 ft-lb. Note the inverse pattern for m_cloud and v_cloud.

PE, ft-lb

    The allowed values for v_lurch are -0.8 to -6.4 ft s^-1. The extreme ends of this range are produced by the highest and lowest m_cloud and are consequently unrealistic. A better range might be -2 to -5 ft s^-1, which is much stronger that the observed. This means either that the head is heavier than 5 lb or that Θ_cl is greater than 30°. Both are probably true.

v_lurch, ft s^-1

For m_head = 6 lb
    The same inverse property of m_cloud and v_cloud is seen here. PE ranges between 110 and 580 ft-lb, like before. As before, v_cloud is not less than 200 ft s^-1.

PE, ft-lb

    The allowed values for v_lurch are -0.8 to -5.7 ft s^-1, very similar to the 5-lb results. A more realistic range would again be -2 to -5 ft s^-1, which is much still stronger than observed.

v_lurch, ft s^-1

For m_head = 7 lb
    Again the inverse pattern between m_cloud and v_cloud.

PE, ft-lb

    The lurch with the 7-lb head is still much too strong, -0.8 to -5.8 ft s^-1 overall and -2 to -5 ft s^-1 for reasonable values. This confirms that 30° is too narrow for Θ_cl.

v_lurch, ft s^-1

For m_head = 8 lb
    The pattern of allowed values for PE is broader vertically and horizontally than with the lighter heads, particularly for small m_cloud and v_cloud. The "smearing" is especially evidence for m_cloud = 0.1 lb. The lowest values of v_cloud now reach down to 100 ft s^-1.

PE, ft-lb

    In spite of the smearing of m_cloud and v_cloud, v_lurch retains its basic range, but finds it shifted toward weaker lurches (-0.2 to -4.7 ft s^-1 overall, -1 to -4 ft s^-1 for reasonable conditions).

v_lurch, ft s^-1

Summary of constraints
    The summary table below shows the full range of allowed values for each of the variables. Three results are that m_cloud has no real constraints, that v_cloud may usually not drop below 200 ft s^-1, and that v_lurch is always negative, includes the observed value by centered above it, and has a range that is nearly independent of m_head.

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