The velocity of a small stone in two similar rivers can be found using the equations below. The equations differ slightly due to the difference in slope between the two rivers, hence resulting in different velocities. It is possible to find the slope at which the two different rivers have equal velocity by solving simultaneously.
\nwhere;
$v$ = velocity of the small stone ($ms^{-1}$)
$g$ = acceleration due to gravity ($9.81ms^{-2}$)
$Q$ = discharge (volume rate of water flow ($m^3s^{-1}$)
$S$ = slope of the river bed
$d_{90}$ = grain size of the small stone ($m$)
You have two unknowns: velocity and slope.
\nYou are trying to work out at which slopes the two rivers have the same velocity. Therefore if you substitute equation one into equation two you are left with an equation where the velocity is equal.
\nRiver one: $v=\\frac{0.37g^{0.33}Q^{0.34}S^{0.20}}{{d_{90}}^{0.35}}$
\nRiver two: $v=\\frac{0.96g^{0.36}Q^{0.29}S^{0.35}}{{d_{90}}^{0.23}}$
\nTherefore: $\\frac{0.37g^{0.33}Q^{0.34}S^{0.20}}{{d_{90}}^{0.35}}=\\frac{0.96g^{0.36}Q^{0.29}S^{0.35}}{{d_{90}}^{0.23}}$
\nNow you only have one unknown: slope.
\nStep one: Multiply both sides by ${d_{90}}^{0.23}$
\n$\\frac{0.37g^{0.33}Q^{0.34}S^{0.20}{d_{90}}^{0.23}}{{d_{90}}^{0.35}}={0.96g^{0.36}Q^{0.29}S^{0.35}}$
\nStep two: Divide both sides by $0.96g^{0.36}Q^{0.29}$
\n$\\frac{0.37g^{0.33}Q^{0.34}S^{0.20}{d_{90}}^{0.23}}{{d_{90}}^{0.35}0.96g^{0.36}Q^{0.29}}= S^{0.35}$
\nStep three: Divide both sides by $S^{0.20}$
\n$\\frac{0.37g^{0.33}Q^{0.34}{d_{90}}^{0.23}}{{d_{90}}^{0.35}0.96g^{0.36}Q^{0.29}}= \\frac{S^{0.35}}{S^{0.20}}$
\nStep four: Cancel to give $S$ alone - $S^{0.35}\\div S^{0.20} = S^{0.15}$
\n$\\frac{0.37g^{0.33}Q^{0.34}{d_{90}}^{0.23}}{{d_{90}}^{0.35}0.96g^{0.36}Q^{0.29}}= S^{0.15}$
\nStep five: Root both sides with respect to $0.15$ to give $S$
\n$\\sqrt[0.15]{\\frac{0.37g^{0.33}Q^{0.34}{d_{90}}^{0.23}}{{d_{90}}^{0.35}0.96g^{0.36}Q^{0.29}}}= S$
\nStep six: Substitute in the given values
\n$\\sqrt[0.15]{\\frac{0.37\\times\\var{g}^{0.33}\\var{Q}^{0.34}{\\var{d90}^{0.23}}}{\\var{d90}^{0.35}0.96\\times\\var{g}^{0.36}\\var{Q}^{0.29}}}= S$
\nStep seven: Calculate to give a value for $S$
\n$\\var{s}= S$
\nStep eight: Round to three significant figures
\n$=\\var{s3f}$
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\nNeed to know indices and logarithm
", "licence": "Creative Commons Attribution 4.0 International"}, "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "showCorrectAnswer": true, "stepsPenalty": 0, "steps": [{"variableReplacementStrategy": "originalfirst", "type": "information", "showFeedbackIcon": true, "marks": 0, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}, "prompt": "You have two unknowns: velocity and slope.
\nYou are trying to work out at which slopes the two rivers have the same velocity. Therefore if you substitute equation one into equation two you are left with an equation where the velocity is equal.
\nRiver one: $v=\\frac{0.37g^{0.33}Q^{0.34}S^{0.20}}{{d_{90}}^{0.35}}$
\nRiver two: $v=\\frac{0.96g^{0.36}Q^{0.29}S^{0.35}}{{d_{90}}^{0.23}}$
\nTherefore: $\\frac{0.37g^{0.33}Q^{0.34}S^{0.20}}{{d_{90}}^{0.35}}=\\frac{0.96g^{0.36}Q^{0.29}S^{0.35}}{{d_{90}}^{0.23}}$
\nNow you only have one unknown: slope.
\nHint: you will need to recall your knowledge of indices for this question.
"}], "gaps": [{"showPrecisionHint": false, "strictPrecision": true, "precisionPartialCredit": 0, "maxValue": "{S}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacements": [], "precision": "3", "allowFractions": false, "minValue": "{S}", "type": "numberentry", "showFeedbackIcon": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "precisionType": "sigfig", "mustBeReducedPC": 0, "precisionMessage": "You have not given your answer to the correct precision.", "scripts": {}, "correctAnswerFraction": false}, {"maxValue": "", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacements": [], "allowFractions": false, "minValue": "", "type": "numberentry", "showFeedbackIcon": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "scripts": {}, "correctAnswerFraction": false}], "variableReplacements": [], "scripts": {}, "prompt": "The two rivers have the same discharge of $\\var{q}m^3s^{-1}$ and in both rivers, the small stone has a grain size of $\\var{d90}m$.
\nWhat slope (greater than zero) gives a velocity of the small stone that is the same in both rivers? Give you answer to 3 significant figures.
\n[[0]]
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