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Leigh and David are playing a game of laser tag'. David has managed to climb aboard a mining cart travelling along a parabolic track, while Leigh has been caught exposed trying to walk along a narrow path on the edge of a cliff. As long as David is in the cart he can only fire his laser at right angles to the direction of motion. If Leigh is caught in David's sight line he is doomed, but he can see a gap between the cliff and some rocks ahead of him. If he can get far enough into the gap David will not be able to see him and tag him. A rough schematic of the situation is sketched below. Cart Tracks David Direction of Fire Rocks Leigh Safety Cliff Edge This situation can be modelled on the cartesian plane as follows, where all quantities are measured in SI units: • Let s e [0, 00) be a variable representing time. Leigh's position is represented by a point that travels along the x-axis, starting at the origin, with a positive constant speed v. • David's position is represented by a point that travels along the parabola y = x² – 6x + 14 so that at time s the x-coordinate is s

Leigh and David are playing a game of laser tag'. David has managed to climb aboard a mining cart travelling along a parabolic track, while Leigh has been caught exposed trying to walk along a narrow path on the edge of a cliff. As long as David is in the cart he can only fire his laser at right angles to the direction of motion. If Leigh is caught in David's sight line he is doomed, but he can see a gap between the cliff and some rocks ahead of him. If he can get far enough into the gap David will not be able to see him and tag him. A rough schematic of the situation is sketched below. Cart Tracks David Direction of Fire Rocks Leigh Safety Cliff Edge This situation can be modelled on the cartesian plane as follows, where all quantities are measured in SI units: • Let s e [0, 00) be a variable representing time. Leigh's position is represented by a point that travels along the x-axis, starting at the origin, with a positive constant speed v. • David's position is represented by a point that travels along the parabola y = x² – 6x + 14 so that at time s the x-coordinate is s

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Leigh and David are playing a game of laser tag'. David has managed to climb aboard a mining cart travelling along a parabolic track, while Leigh has been caught exposed trying to walk along a narrow path on the edge of a cliff. As long as David is in the cart he can only fire his laser at right angles to the direction of motion. If Leigh is caught in David's sight line he is doomed, but he can see a gap between the cliff and some rocks ahead of him. If he can get far enough into the gap David will not be able to see him and tag him. A rough schematic of the situation is sketched below. Cart Tracks David Direction of Fire Rocks Leigh Safety Cliff Edge This situation can be modelled on the cartesian plane as follows, where all quantities are measured in SI units: • Let s e [0, 00) be a variable representing time. • Leigh's position is represented by a point that travels along the x-axis, starting at the origin, with a positive constant speed v. • David's position is represented by a point that travels along the parabola y = x² – 6x + 14 so • that at time s the x-coordinate is s. ser tag is a game where you try to shoot (tag) your opponent using a low-powered laser
Transcribed Image Text:Leigh and David are playing a game of laser tag'. David has managed to climb aboard a mining cart travelling along a parabolic track, while Leigh has been caught exposed trying to walk along a narrow path on the edge of a cliff. As long as David is in the cart he can only fire his laser at right angles to the direction of motion. If Leigh is caught in David's sight line he is doomed, but he can see a gap between the cliff and some rocks ahead of him. If he can get far enough into the gap David will not be able to see him and tag him. A rough schematic of the situation is sketched below. Cart Tracks David Direction of Fire Rocks Leigh Safety Cliff Edge This situation can be modelled on the cartesian plane as follows, where all quantities are measured in SI units: • Let s e [0, 00) be a variable representing time. • Leigh's position is represented by a point that travels along the x-axis, starting at the origin, with a positive constant speed v. • David's position is represented by a point that travels along the parabola y = x² – 6x + 14 so • that at time s the x-coordinate is s. ser tag is a game where you try to shoot (tag) your opponent using a low-powered laser
• The rocks are represented by a the line segment r = 10 for y 2 3. (a) Determine the (x, y) co-ordinates for Leigh's and David's position at time s. (b) Find the equation of a normal line to the parabola, in terms of s, which passes through David's position. (c) Find the point H where the normal line found in (b) intersects the r-axis in terms of s. (d) If there were no rocks available for Leigh to hide in, would he be able to escape from David without being tagged? Justify your answer.
Transcribed Image Text:• The rocks are represented by a the line segment r = 10 for y 2 3. (a) Determine the (x, y) co-ordinates for Leigh's and David's position at time s. (b) Find the equation of a normal line to the parabola, in terms of s, which passes through David's position. (c) Find the point H where the normal line found in (b) intersects the r-axis in terms of s. (d) If there were no rocks available for Leigh to hide in, would he be able to escape from David without being tagged? Justify your answer.
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