Difference between revisions of "Piece-wise defined functions"
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Alice is doing bungee jumping and jumps from a tall bridge. She uses a bungee cord that is 10 meters long (non-stretched). The spring constant of the cord is 500 N/m. Alice weighs 60 kg. Find how Alices velocity changes as a function of distance from the bridge as she descents form the bridge to the lowest point. You can make necessary simplifying assumption (no air resistance, weight of the cord is negligible, Alice jumps directly downwards, etc.). | Alice is doing bungee jumping and jumps from a tall bridge. She uses a bungee cord that is 10 meters long (non-stretched). The spring constant of the cord is 500 N/m. Alice weighs 60 kg. Find how Alices velocity changes as a function of distance from the bridge as she descents form the bridge to the lowest point. You can make necessary simplifying assumption (no air resistance, weight of the cord is negligible, Alice jumps directly downwards, etc.). | ||
| − | Choose the x-axis (in meters) pointing downwards and with the origin coinciding with Alice's initial position on the bridge. For the first part of her journey, i.e. when <math> 0\le x \le 10 </math>, the only acting force is gravity, so the speed is <math> gt </math>, or <math> \sqrt{2gx} </math>. After the point where <math> x = 10 </math>, a restoring force comes into play | + | Choose the x-axis (in meters) pointing downwards and with the origin coinciding with Alice's initial position on the bridge. For the first part of her journey, i.e. when <math> 0\le x \le 10 </math>, the only acting force is gravity, so the speed is <math> gt </math>, or <math> v(x) = \sqrt{2gx} </math>. After the point where <math> x = 10 </math>, a restoring force comes into play. Velocity at that point is <math> v(10) = \sqrt{2g\cdot 10} = 2\sqrt{5g} </math>. Therefore velocity can be computed using |
''' Ball trajectory''' | ''' Ball trajectory''' | ||
A person throws a ball forward at 3 m/s. The ball bounces off the ground, reaches some maximal height, then falls and bounces again, and so on. If the initial altitude of the ball is 1.5 m, find the trajectory of the ball if a) no energy is lost b) about 20% of energy is lost on each bounce. | A person throws a ball forward at 3 m/s. The ball bounces off the ground, reaches some maximal height, then falls and bounces again, and so on. If the initial altitude of the ball is 1.5 m, find the trajectory of the ball if a) no energy is lost b) about 20% of energy is lost on each bounce. | ||
Revision as of 20:39, 3 December 2021
A typical introduction of piece-wise defined functions in a calculus textbook looks something like this: there are functions that take on different formulas depending on the intervals (pieces) of their domains, for example: [math]\displaystyle{ f(n) = \begin{cases} x^2, & \mbox{if }x\le 1 \\ 3-2x, & \mbox{if }x \gt 1 \end{cases} }[/math]
(examples may vary).
Below we try to give a bit more motivation to piece-wise functions (which is not just a nuance for students, but rather indispensable tool in mathematics and applications).
Pool table
A pool table has dimensions 10 ft by 5 ft. A ball is hit from a corner towards the longer side at the angle 30 degrees (with respect to the shorter side). Describe the trajectory of the ball before it hits the side second time.
Even numbers
Determine how many positive even numbers are there thad do not exceed a given positive integer.
Bungee jumping
Alice is doing bungee jumping and jumps from a tall bridge. She uses a bungee cord that is 10 meters long (non-stretched). The spring constant of the cord is 500 N/m. Alice weighs 60 kg. Find how Alices velocity changes as a function of distance from the bridge as she descents form the bridge to the lowest point. You can make necessary simplifying assumption (no air resistance, weight of the cord is negligible, Alice jumps directly downwards, etc.).
Choose the x-axis (in meters) pointing downwards and with the origin coinciding with Alice's initial position on the bridge. For the first part of her journey, i.e. when [math]\displaystyle{ 0\le x \le 10 }[/math], the only acting force is gravity, so the speed is [math]\displaystyle{ gt }[/math], or [math]\displaystyle{ v(x) = \sqrt{2gx} }[/math]. After the point where [math]\displaystyle{ x = 10 }[/math], a restoring force comes into play. Velocity at that point is [math]\displaystyle{ v(10) = \sqrt{2g\cdot 10} = 2\sqrt{5g} }[/math]. Therefore velocity can be computed using
Ball trajectory
A person throws a ball forward at 3 m/s. The ball bounces off the ground, reaches some maximal height, then falls and bounces again, and so on. If the initial altitude of the ball is 1.5 m, find the trajectory of the ball if a) no energy is lost b) about 20% of energy is lost on each bounce.