Related rates with point moving along a curve or two moving objects

Questions where a point moves along a given curve and rates of change of coordinates are related, or two moving objects approach a common point and the rate of change of distance between them is required.

2 questions · Standard +0.8

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CAIE P1 2016 June Q7
7 marks Standard +0.8
7 The point \(P ( x , y )\) is moving along the curve \(y = x ^ { 2 } - \frac { 10 } { 3 } x ^ { \frac { 3 } { 2 } } + 5 x\) in such a way that the rate of change of \(y\) is constant. Find the values of \(x\) at the points at which the rate of change of \(x\) is equal to half the rate of change of \(y\).
Pre-U Pre-U 9795/2 2013 November Q7
Standard +0.8
7 At a given instant two stunt cars, \(X\) and \(Y\), are at distances 500 m and 800 m respectively from the point of intersection, \(O\), of two straight roads crossing at right angles. The stunt cars are approaching \(O\) at uniform speeds of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively, one on each road. Find, in either order,
  1. the time taken to reach the point of closest approach,
  2. the shortest distance between the stunt cars.