What is the Speed of a Book Sliding Down an Inclined Plane?

In summary, the problem involves a book sliding down an inclined plane with an initial velocity of 0 m/s and a final velocity of 2.5 m/s at the bottom of the slope. The question asks for the book's speed at half way and a third way down the slope. To solve this, the standard constant acceleration equations are used, with the distance being represented as "d". By equating two equations and solving for the final velocity, the book's speed at half way and a third way down the slope can be determined. There is also a mention of using conservation of energy to solve the problem, but it is noted that friction should be accounted for in this approach.
  • #1
The_big_dill
33
0

Homework Statement


A Book (initially at rest) slides down an inclined plane, when it is at the bottom of the slope, its speed is 2.5 m/s. What is its speed half way and a third way down the slope.

V1 = 0
t1 = 0
V2 = 2.5 m/s

Homework Equations



a = [tex]\Delta[/tex]v / [tex]\Delta[/tex] t

v = [tex]\Delta[/tex]d / [tex]\Delta[/tex] t

Any of the other Uniform motion equations (kinematics)

The Attempt at a Solution



It looks simple, but i simply cannot figure out where to start...

I have to many missing variables, which unless i can assume something or equate 2 equations, i have no idea what to do.

I know for a fact that i am not suppose to be using forces (ex. x-component of acceleration due to gravity), i should be using kinematics.
 
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  • #2
Hi The_big_dill! :smile:

(have a delta: ∆ :wink:)
The_big_dill said:
A Book (initially at rest) slides down an inclined plane, when it is at the bottom of the slope, its speed is 2.5 m/s. What is its speed half way and a third way down the slope.

Use the standard constant acceleration equations.

You have vi vf and s, and you want to find vf for a different s …

what equations do you think you should use? :smile:
 
  • #3
Also, becareful which axis you use.

If you want to take the book's direction of travel to be the x-axis, then you need to ressolve for g.

If you wish to take g as the y-axis, then you need to ressolve for the x and y components of the sliding motion.

After you've decided on your axis, apply your equations of motion as suggested ^^
 
  • #4
Hi Ambidext! :smile:

No, the question doesn't give the angle.

This is a purely 1D problem, not needing g, in which we could work out the angle from the given data, but we're not asked for it. :wink:
 
  • #5
tiny-tim said:
Hi Ambidext! :smile:

No, the question doesn't give the angle.

This is a purely 1D problem, not needing g, in which we could work out the angle from the given data, but we're not asked for it. :wink:

Exactly, thank you for clarifying that XD

Also, tiny tim, i am either not familiar with that equation, or i already listed it in my original post.

If you mean a = [tex]\Delta[/tex]V/[tex]\Delta[/tex]t, then i don't have [tex]\Delta[/tex]t, i only have t1... which is 0... which is undefined.
 
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  • #6
(What happened to that ∆I gave you? :confused:)

There are three standard constant acceleration equations … look them up. :wink:

Then call the distance "s" (and later s/2 and s/3), and carry on from there. :smile:
 
  • #7
I think there is some confusion here...

I use "d" to represent distance

V1 to represent initial velocity

V2 to represent final velocity

The only other equation that makes sense to me (but still missing multiple variables):

V22 = V12 + 2a[tex]\Delta[/tex]d

I don't have distance, i also don't have acceleration...
 
  • #8
ok, then call the distance "d" (and later d/2 and d/3), and carry on from there
 
  • #9
tiny-tim said:
ok, then call the distance "d" (and later d/2 and d/3), and carry on from there

I cannot solve for d. I do not have 'a', or delta 't'.

Maybe i am missing something, but i do not see it.
 
  • #10
What equations do you know involving d and t and only one v ?

(If you don't know any, look it up)
 
  • #11
tiny-tim said:
What equations do you know involving d and t and only one v ?

(If you don't know any, look it up)

I do know 2 that are very similar.

[tex]\Delta[/tex]d = V1[tex]\Delta[/tex]t + 1/2a[tex]\Delta[/tex]t2

[tex]\Delta[/tex]d = V2[tex]\Delta[/tex]t - 1/2a[tex]\Delta[/tex]t2

[STRIKE]Hmmm, are you suggesting to equate them?
[/STRIKE]
[STRIKE]I should end up with a quadratic.[/STRIKE]
 
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  • #12
oops!

Sorry, I forgot what the question was :redface:

we're looking for vf, not t, so we need an equation involving d and two vs (but no t).
 
  • #13


tiny-tim said:
Sorry, I forgot what the question was :redface:

we're looking for vf, not t, so we need an equation involving d and two vs (but no t).

V22 = V12 + 2ad
 
  • #14
Yes. :smile:

Go on, then. :wink:
 
  • #15
tiny-tim said:
Yes. :smile:

Go on, then. :wink:

... I am missing 2 variables, d and a.

2.52 = 02 + 2a[tex]\Delta[/tex]d
 
  • #16
rl.bhat said:
For Δd/2, the equation becomes,

v22 = 2aΔd/2 ...(2)

Divide eq.(1) by (2), and solve for v2

Interesting... I will give that a go.

I already know V2.
 
  • #17


The_big_dill said:
V22 = V12 + 2ad...(1)

Write two more equations for d/2 and d/3.
To eliminate unknown quantities, divide eq.(1) by eq.(2) and (3) and solve for required velocities.
 
  • #18
What's the problem with just using conservation of Energy.?
 
  • #19


rl.bhat said:
Write two more equations for d/2 and d/3.
To eliminate unknown quantities, divide eq.(1) by eq.(2) and (3) and solve for required velocities.

I already have velocity 1 and 2... i need distance, time or acceleration.
 
  • #20
SammyS said:
What's the problem with just using conservation of Energy.?

I do not know how you are going to solve it without any angles.
 
  • #21
The_big_dill said:
I do not know how you are going to solve it without any angles.
You just need the relative heights. I assume the incline has a constant slope!

So at half way down the incline, h1/2 =(1/2) h, where h is the initial height.

And h1/3 =(2/3) h.

mgh‒mgh(1/2) = (1/2)mgh.

mgh‒mgh(1/3) = (1/3)mgh.

From the given information, mgh = (1/2)m(2.5)^2, in units of Joules.

Added in Edit:

Well, I should have accounted for friction. The energy dissipated by friction is proportional to the distance down the incline which is proportional to Δh, so it works out the same.

Looks like nobody else has even mentioned friction.
 
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1. How does the angle of the slope affect the speed of the book?

The steeper the slope, the faster the book will slide down due to the increased force of gravity pulling it downward.

2. What is the role of friction in book sliding on a slope?

Friction acts as a force that opposes the motion of the book, causing it to slow down and eventually come to a stop.

3. Can the mass of the book affect its sliding speed on a slope?

Yes, the greater the mass of the book, the more force is required to overcome its inertia and accelerate it down the slope.

4. Does the surface of the slope impact the book's sliding speed?

Yes, a smoother surface will result in less friction and a faster sliding speed, while a rougher surface will increase friction and slow down the book's motion.

5. How can we calculate the acceleration of the book on a slope?

The acceleration of the book can be calculated using the formula a = gsinθ, where g is the acceleration due to gravity (9.8 m/s²) and θ is the angle of the slope.

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