Simple particle on slope confusion

• rmawatson
In summary, you are trying to find the velocity of a particle at a given location, but you get different answers depending on the equation used.
rmawatson

Homework Statement

a particle on a slope with angle theta with no friction. v(0) = 0, x(0) = 0, with coordinates i down the slope and j normal to it.

I am confused about why with the constant velocity forumla I get a different answer to my attempted method.. I can't see what's wrong..I need to find the velocity at "l"
x is the top of the slope of a particle on a smooth surface, with no friction,
v(0) = 0, x(0) = 0

along i direction I am starting with:

mgsin(theta) == ma

gsin(theta) == a

using constant acceleration formula

v^2 = v0^2 + 2a0(x-x0)
v^2 = 0 + 2gsin(theta)(x-0)
v = sqrt( 2gl*sin(theta) )

My original attempt below is wrong, but I can't see why. I want to know what it doesn't work the same.

so from
gsin(theta) == a

Integrating wrt t

gsin(theta)t == v + c

Integrating wrt t again

1/2*gsin(theta)t^2 == x + ct + d

with v(0) == 0 and x(0) == 0

0 = c and d = 0

so if I now plugged in 'l' to the equation for position

1/2*gsin(theta)t^2 == l

and solve for t I get,

t = sqrt[ (2l)/(gsin(theta)) ]

so this is the time at which position == l ?

If I then plug this time into the equation for velocity,

gsin(theta)t == v

gsin(theta)*sqrt[ (2l)/(gsin(theta)) ] = v

not the same as with the constant velocity forumla..
why ? what is wrong with this method

Thanks for any help

rmawatson said:

Homework Statement

a particle on a slope with angle theta with no friction. v(0) = 0, x(0) = 0, with coordinates i down the slope and j normal to it.

I am confused about why with the constant velocity forumla I get a different answer to my attempted method.. I can't see what's wrong..I need to find the velocity at "l"
x is the top of the slope of a particle on a smooth surface, with no friction,
v(0) = 0, x(0) = 0

along i direction I am starting with:

mgsin(theta) == ma

gsin(theta) == a

using constant acceleration formula

v^2 = v0^2 + 2a0(x-x0)
v^2 = 0 + 2gsin(theta)(x-0)
v = sqrt( 2gl*sin(theta) )

My original attempt below is wrong, but I can't see why. I want to know what it doesn't work the same.

so from
gsin(theta) == a

Integrating wrt t

gsin(theta)t == v + c

Integrating wrt t again

1/2*gsin(theta)t^2 == x + ct + d

with v(0) == 0 and x(0) == 0

0 = c and d = 0

so if I now plugged in 'l' to the equation for position

1/2*gsin(theta)t^2 == l

and solve for t I get,

t = sqrt[ (2l)/(gsin(theta)) ]

so this is the time at which position == l ?

If I then plug this time into the equation for velocity,

gsin(theta)t == v

gsin(theta)*sqrt[ (2l)/(gsin(theta)) ] = v

not the same as with the constant velocity forumla..
why ? what is wrong with this method

Hello rmawatson. Welcome to PF!

Those answers are the same !

If I plug in numbers I get a different answer for both??

rmawatson said:
If I plug in numbers I get a different answer for both??
Example ... ?

Looks like I did something wrong when I checked it. I was so sure that my formula must be wrong (as it was a different way to the book and done by me) I didn't check twice.

You are right, and with some simple rearranging it comes out the same... thank you for your help.

1. What is a simple particle on slope?

A simple particle on slope is a theoretical model used in physics to understand the motion of a single particle on a sloped surface. It assumes that the particle is moving in a straight line and is subject to the forces of gravity and friction.

2. How does the slope affect the motion of a simple particle?

The slope of the surface will determine the direction and speed of the particle's motion. A steeper slope will result in a faster acceleration due to the greater force of gravity, while a gentler slope will result in a slower acceleration.

3. What is the role of gravity in the motion of a simple particle on slope?

Gravity is the force that pulls the particle towards the center of the Earth, causing it to accelerate down the slope. Without gravity, the particle would not move at all.

4. How does friction affect a simple particle on slope?

Friction is a force that acts in the opposite direction of motion and is caused by the contact between the particle and the surface. Friction will slow down the particle's motion and ultimately bring it to a stop.

5. What are some real-world applications of the simple particle on slope model?

The simple particle on slope model is used in various fields such as engineering, geology, and sports to understand and predict the motion of objects on inclined surfaces. For example, it can be used to design roller coasters, analyze landslides, and predict the trajectory of a ball rolling down a hill.

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