Parametric equations motion problem

In summary: Could you point me to a page with identities like that (preferably as simple as possible)?I am not aware of a simpler form. The substitutionu = 4 sinθ - 3 cosθleads tosinθ - 4/5 = u/5cosθ - 3/5 = √(1 - u²)/5and substituting those into the identitysinθ cosψ - cosθ sinψ = Agivesu/5 - √(1 - u²)/5 = Awhich simplifies to-u² + 5u + 24 = 0which can be solved for u using the quadratic formula, and then solving
  • #1
cjh12398
6
0
The question states:
Two towns A and B are located directly opposite each other on a river 8km wide which flows at a speed 4km/h. A person from town A wants to travel to a town C located 6km up-stream from and on the same side as B. The person travels in a boat with maximum speed 10km/h and wishes to reach C in the shortest possible time. Let x(t) be the distance traveled upstream and y(t) be the distance traveled across the river in t hours. The person heads out at angle theta.

a) Show that x(t)=10tcos(theta)-4t and y(t)=10tsin(theta)
b) What is the angle theta and how long would the trip take?

Relevant equations:
So far I have used v=d/t along with some vector diagrams.

My attempt:
I have proven a) already by using v=d/t. The net velocity for x was equal to 10cos(theta)-4 and I just rearranged for x. I did the same to find y.

I then found the angle theta by saying that sin(theta)=8/10, therefore theta=arcsin(4/5). Also, I found the theta in terms of arccos which was theta=arccos(3/5). I found these by using a distance triangle with adjacent=6, opposite=8 and hypotenuse=10.

I then equated x(t)=6 ==> 10tcos(theta)-4t=6
10tcos(arccos(3/5))-4t=6
10t(3/5)-4t=6
6t-4t=6
t=3
And equated y(t)=8 ==> 10tsin(theta)=8
10tsin(arcsin(4/5))=8
10t(4/5)=8
8t=8
t=1
This is where I'm having problems. Shouldn't the time value be equal? If anyone could please help me out I would greatly appreciate it.
 
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  • #2
welcome to pf!

hi cjh12398! welcome to pf! :smile:
cjh12398 said:
… The person heads out at angle theta.

a) Show that x(t)=10tcos(theta)-4t and y(t)=10tsin(theta)
b) What is the angle theta and how long would the trip take?

I then found the angle theta by saying that sin(theta)=8/10,

no, the question means that the boat is heading at angle θ relative to the water, not to the land

use your two equations in a) to find θ (by eliminating t)​
 
  • #3
Thank you, I continued working out the times for both x and y after using theta by eliminating the t, but I had to use a graphics calculator to solve for theta.
The equation I got to solve theta was:
sin(theta)-(4/3)cos(theta)+(8/15)=0.
In my course it is calculator free, so is possible to solve this by hand? I've tried for about 15 mins...
 
  • #4
try to re-arrange it so that it's of the form sinθcosψ - cosθsinψ = A,

then use one of the standard trigonometric identities :wink:
 
  • #5
cjh12398 said:
I then found the angle theta by saying that sin(theta)=8/10
You left out the [itex]t[/itex]. Your equation was [itex]y(t) = 10\; t \sin\theta[/itex].

It would help if you first solved this (and all other) problems symbolically, and only plugged in numbers as the very last step. It would also help if you put spaces in your equations; they don't cost anything, and they let the various terms and factors stand out more clearly.
 
  • #6
I'm also stuck with this one.

I hit the same brick wall by myself with the trig identities, been trying for about 2 hours now to find a way to do this question :(

Treating it as a vector addition from a point before the river flow is considered is something I will try.
 
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  • #7
tiny-tim said:
try to re-arrange it so that it's of the form sinθcosψ - cosθsinψ = A,

then use one of the standard trigonometric identities :wink:

I'm sorry I can't get it at all near that form. Using the Asin(x) + Bcos(x), i can solve for theta with a correct answer.
 
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  • #8
duldin said:
I'm sorry I can't get it at all near that form. Using the Asin(x) + Bcos(x) identity on that page I've not seen before …

that is that form (with B/A = -tanψ) :wink:
… I have an arctan(-4/3) term which gives the same issue of not being able to do by hand...

i don't think they'd expect you to do it by hand

change it to arcsin(-4/5), and use sin tables or your calculator :smile:
 
  • #9
Aye, I've worked through it all and trying to forget about it now ;). It didn't occur to me originally, but eventually I realized to just leave it as the difference of some inverse trig functions. Messy but it does appear to be the best I can do... substituting theta back in even more messy :P

Thanks for your help. I'd not seen that trig equation before.
 

1. What is a parametric equation and how is it used in motion problems?

A parametric equation is a set of equations that express the values of a set of variables as functions of one or more independent variables, typically represented by t. In the context of motion problems, parametric equations are used to describe the position, velocity, and acceleration of an object as a function of time.

2. How do you find the parametric equations for a motion problem?

In order to find the parametric equations for a motion problem, you first need to identify the independent variable (usually time) and the dependent variables (position, velocity, and acceleration). Then, you can use the given information and equations of motion (such as displacement = initial velocity * time + 0.5 * acceleration * time^2) to write the parametric equations.

3. Can parametric equations be used to solve complex motion problems?

Yes, parametric equations can be used to solve complex motion problems that involve multiple objects or changing conditions. By incorporating different equations for position, velocity, and acceleration, parametric equations allow for a more comprehensive and accurate description of the motion.

4. How do you graph parametric equations for a motion problem?

To graph parametric equations for a motion problem, you can use a graphing calculator or online graphing tool. First, enter the equations for position, velocity, and acceleration as functions of time. Then, set the range for the independent variable (usually time) and the corresponding units. The resulting graph will show the position, velocity, and acceleration of the object over time.

5. Can parametric equations be applied to real-world motion problems?

Yes, parametric equations are commonly used in real-world motion problems, such as in physics, engineering, and sports. They allow for a more precise and detailed analysis of an object's motion, taking into account factors such as initial conditions, acceleration, and varying forces.

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