Block Jump from Ramp: Find Speed v

[racecar12]

Homework Statement

A block starts at rest and slides down a frictionless track except for a small rough area on a horizontal section of the track
It leaves the track horizontally, flies through the air, and subsequently strikes the ground. The acceleration of gravity is 9.8 m/s2 .

What is the speed v of the block when it leaves the track?

Givens:
mass of block =0.401kg
Starting height of block = 4.1m
Horizontal height of block when it encounters the friction = 1.9m
Length of the rough area on the horizontal section of track = 1.3m
Coefficient of friction = 0.3

Homework Equations

E_initial=E_final
K_initial +U_initial = K_final +U_final

F_friction=μN
N=W=mg

Δy=h=4.1-1.9=2.2

The Attempt at a Solution

I set K_initial and U_final to zero, yielding:
mgh=0.5mv²d - F_frictiond
v_final= [ $gh+μNd/0.5(1.3)$ ]^1/2
v=5.96m/s

Just wondering if I went about this correctly,
Thanks

 

[tiny-tim]

hi racecar12!

mgh=0.5mv²d - F_frictiond
v_final= [ $gh+μNd/0.5(1.3)$ ]^1/2

i'm confused …

can you check that, and write it out again, please​

 

[racecar12]

Thanks for looking at this post.

Here was my though process as I approached this problem...

We'll just use the conservation of energy theorm for this problem.

E_initial=E_final
K_initial+U_initial=K_final+U_final

Since the massive object has no initial kinetic energy at the start of the system, call it when the object is at its initial height, it has potential energy equal to mgh. m is the mass of the object, g is the acceleration due to gravity, and h is equal to the distance the object travels. In this case, I have to set h equal to a change in the initial height if the object to its height when it encounters the frictional surface. Δy=h. The left side of the equation now reads: U_initial=mgh. Or just, mgh

mgh=some final kinetic energy - the work done by friction
We have an equation that relates kinetic energy with velocity...it is 0.5mv²=K_final...v² is also the final velocity, which for this system, is right at the point when object leaves the surface. The object will travel in the air for some unknown distance at some other velocity, but for purposes of this problem, we won't worry about them, because those things are not what this problem is asking for. Now, I'm not sure about multiplying, by length d, but I believe we have to.

I also know that the work done by the frictional force equals μNd. That's because work equals force x distance. We have μ and d given to us. d in this case is the distance the object travels along the rough surface only. N is unknown, but we do know that in this case N =W, which happens to = mg in this case. I may not have told you that the rough surface is flat. The minus sign comes in because the frictional force opposes the motion of the object.

Now our equation looks like this
mgh = 0.5mv²d - 1.532622

Now let's isolate v. My m will cancel when I move everything over to the other side.

(gh + 1.532622)/(0.5)(d) =v²
PlUg it all in and take the square root...v = 5.96 m/s

And sorry that the + is underlined..I'm not sure why it does that when I type it in.

 

[tiny-tim]

We have an equation that relates kinetic energy with velocity...it is 0.5mv²=K_final...v² is also the final velocity, which for this system, is right at the point when object leaves the surface. The object will travel in the air for some unknown distance at some other velocity, but for purposes of this problem, we won't worry about them, because those things are not what this problem is asking for. Now, I'm not sure about multiplying, by length d, but I believe we have to.

ah, that's what was puzzling me

no, you don't multiply by d (i've no idea twhere you got that idea from)

KE = 1/2 mv², that's all!

try again ​

 

[racecar12]

Yes, you are correct that kinetic energy equals 0.5mv². However, removing d from the equation takes me much farther from the answer, which I know to be 5.95617m/s. if I proceed with the way that I have thought, I get v to equal 5.96046. Close, but no cigar.

I am making a mistake somewhere in my thought process, but I don't know where it is. Perhaps I'm making multiple mistakes, but I don't know where or what they are. Thanks

 

[haruspex]

Yes, you are correct that kinetic energy equals 0.5mv². However, removing d from the equation takes me much farther from the answer, which I know to be 5.95617m/s. if I proceed with the way that I have thought, I get v to equal 5.96046. Close, but no cigar.

I am making a mistake somewhere in my thought process, but I don't know where it is. Perhaps I'm making multiple mistakes, but I don't know where or what they are. Thanks

Can't tell where your error is unless you post the full details of your calculation.

 

[racecar12]

Can't tell where your error is unless you post the full details of your calculation.

Seriously?

Thanks for looking at this post.

Here was my though process as I approached this problem...

We'll just use the conservation of energy theorm for this problem.

E_initial=E_final
K_initial+U_initial=K_final+U_final

Since the massive object has no initial kinetic energy at the start of the system, call it when the object is at its initial height, it has potential energy equal to mgh. m is the mass of the object, g is the acceleration due to gravity, and h is equal to the distance the object travels. In this case, I have to set h equal to a change in the initial height if the object to its height when it encounters the frictional surface. Δy=h. The left side of the equation now reads: U_initial=mgh. Or just, mgh

mgh=some final kinetic energy - the work done by friction
We have an equation that relates kinetic energy with velocity...it is 0.5mv²=K_final...v² is also the final velocity, which for this system, is right at the point when object leaves the surface. The object will travel in the air for some unknown distance at some other velocity, but for purposes of this problem, we won't worry about them, because those things are not what this problem is asking for. Now, I'm not sure about multiplying, by length d, but I believe we have to.

I also know that the work done by the frictional force equals μNd. That's because work equals force x distance. We have μ and d given to us. d in this case is the distance the object travels along the rough surface only. N is unknown, but we do know that in this case N =W, which happens to = mg in this case. I may not have told you that the rough surface is flat. The minus sign comes in because the frictional force opposes the motion of the object.

Now our equation looks like this
mgh = 0.5mv²d - 1.532622

Now let's isolate v. My m will cancel when I move everything over to the other side.

(gh + 1.532622)/(0.5)(d) =v²
PlUg it all in and take the square root...v = 5.96 m/s

And sorry that the + is underlined..I'm not sure why it does that when I type it in.

 

[kris2fer]

Now our equation looks like this
mgh = 0.5mv²d - 1.532622

You don't need the "d" term and try adding the 1.5 term instead of subtracting since there's thermal energy at the bottom of the ramp.

 

[racecar12]

You don't need the "d" term and try adding the 1.5 term instead of subtracting since there's thermal energy at the bottom of the ramp.

Hey, thanks for your constructive contribution to my inquiry. I attempted the solution in the way you suggested. I get v to equal 6.3228, which is unfortunately not the correct answer.

I'm wondering if I may have to consider the total length the object travels...like adding Δy to the length of the rough surface.

 

[kris2fer]

I get v to equal 6.3228, which is unfortunately not the correct answer.

Really? The equation i get is mgh = 0.5mv² + umgd
Solve for v and you should get about 5.956173268m/s.

 

[racecar12]

Oh man, you are correct. I made an algebra mistake while moving everything over to solve for v. Good catch. And thanks a bunch!

 

[haruspex]

Can't tell where your error is unless you post the full details of your calculation.

Seriously?

It had been pointed out to you that mv²d was wrong. You replied that you tried taking out the d and it made matters worse, but you didn't post the calculation. There's also the matter of where 1.532622 came from. I don't see an explanation of that.

 

[tiny-tim]

(just got up :zzz:)

Seriously?

yes, seriously …

I made an algebra mistake while moving everything over to solve for v.

that is why we usually insist on you showing your full calculations

please do so next time, and save yourself some time! ​