Parametric equations for a particle

In summary, the parametric equations x = t +2/t and y = 3t^2 describe the path of a particle for t > 0. The velocity of the particle in the x direction is zero when t = sqrt(1/2) and the coordinates of this point are (3.5355, 1.5). When t =
  • #1
rjs123
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0

Homework Statement




The path of a particle is given for time t > 0 by the parametric equations: x = t +2/t, y = 3t^2

a. Find the coordinates of each point on the path where the velocity of the particle in the x direction is zero.

b. Find dy/dx when t = 1

c. Find d^2y/dx^2 when y = 12



The Attempt at a Solution



a. No horizontal tangents...dy/dt = 6t...= 0 when t = 0...but t must be greater than 0 (condition)

b. dy/dx = (dy/dt)/(dx/dt) = 6t/(1-2/t^2) = -6...when you plug in 1

c. stuck on c at the moment
 
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  • #2
For a) you want to get were dx/dt = 0 and then rearrange for t. This will give a value for t, so you can get the x and y coordinates for it.

b) is correct.

For part c, differentiate the expression wrt t using implicit differentiation, can you do this?
 
  • #3
a. dx/dt = 1 - 2/t^2 = 0

1 = 2/t^2

1/2 = t^2

t = sqrt(1/2)


x = 3.5355, y = 1.5


this is what i got for your recommendation.
 
Last edited:
  • #4
rjs123 said:
a. dx/dt = 1 - 2/t^2 = 0

1 = 2/t^2

1/2 = t^2

t = sqrt(1/2)


x = 3.5355, y = 1.5


this is what i got for your recommendation.

I thought when the velocity of the particle in the x direction is zero when there is a horizontal tangent present?

The gradient function of x is dx/dt. So if dx/dt = 0 (or the velocity in the x-direction is zero), then the tangent drawn has a zero gradient i.e. is a horizontal tangent.
 
  • #5
rock.freak667 said:
The gradient function of x is dx/dt. So if dx/dt = 0 (or the velocity in the x-direction is zero), then the tangent drawn has a zero gradient i.e. is a horizontal tangent.

yes, the graph has no points where there could be a horizontal tangent line...the one found on your recommendation contains a slope of -2...at (3.5, 1.5)
 
  • #6
rjs123 said:
yes, the graph has no points where there could be a horizontal tangent line...the one found on your recommendation contains a slope of -2...at (3.5, 1.5)

if x = t + 2/t had no stationary points then putting dx/dt = 0 would lead to something like 0=2.

However there would be a slope since the vertical velocity is not zero at t = 1.
 
  • #7
rjs123 said:

Homework Statement




The path of a particle is given for time t > 0 by the parametric equations: x = t +2/t, y = 3t^2
Is x= t+ (2/t) or (t+2)/t?

a. Find the coordinates of each point on the path where the velocity of the particle in the x direction is zero.

b. Find dy/dx when t = 1

c. Find d^2y/dx^2 when y = 12



The Attempt at a Solution



a. No horizontal tangents...dy/dt = 6t...= 0 when t = 0...but t must be greater than 0 (condition)

b. dy/dx = (dy/dt)/(dx/dt) = 6t/(1-2/t^2) = -6...when you plug in 1

c. stuck on c at the moment
 
  • #8
For (a): If the velocity in the x direction is zero, then the tangent to the path is vertical, not horizontal.

For (c):
[tex]\frac{d^2y}{dx^2}=\frac{d\left(dy/dx\right)}{dx}=\frac{d\left(dy/dx\right)/dt}{dx/dt}[/tex]​
and you found that
[tex]\frac{dy}{dx}=\frac{6t}{1-2t^{-2}}[/tex]​
 

1. What are parametric equations for a particle?

Parametric equations for a particle are mathematical expressions that describe the position, velocity, and acceleration of a particle in motion at a given time, using one or more independent variables.

2. How are parametric equations different from standard equations?

Parametric equations use independent variables, typically represented by the letter t, to describe the position of a particle at a given time, while standard equations use dependent variables, typically represented by x and y, to describe the relationship between two variables.

3. What is the purpose of using parametric equations for a particle?

The purpose of using parametric equations for a particle is to accurately and efficiently describe the motion of a particle in a specific direction and at a specific time.

4. Can parametric equations be used for particles in three-dimensional space?

Yes, parametric equations can be used to describe the motion of particles in three-dimensional space by using three independent variables, typically represented by t, x, and y, to describe the particle's position at a given time.

5. How are parametric equations used in real-world applications?

Parametric equations for a particle are commonly used in physics, engineering, and other scientific fields to model the motion of objects such as projectiles, planets, and vehicles. They are also used in computer graphics to create animations and special effects.

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