Prove This, please -- particle moving with a uniform acceleration

[VeeFlamesPhys]

Homework Statement

Hi all. Please help me with this. A small particle moving with a uniform acceleration a covers a distances X and Y in the first two equal and consecutive intervals of time t. Show that a = Y-X/t².

Homework Equations

Acceleration is usually v-u/t i.e. rate of change of velocity with time.

The Attempt at a Solution

Since Y and X are distances, and t is time, and the unit for acceleration is m/s², I put two and two together. That was pretty easy, but I looked at the question properly and saw there was more to it than that, for instance, the 'first two equal and consecutive intervals of time t' I used different equations of motion such as s=ut+1/2at and v²=u²+2aS, but I kept getting stuck. Help, please!

 

[Dick]

Homework Statement

Hi all. Please help me with this. A small particle moving with a uniform acceleration a covers a distances X and Y in the first two equal and consecutive intervals of time t. Show that a = Y-X/t².

Homework Equations

Acceleration is usually v-u/t i.e. rate of change of velocity with time.

The Attempt at a Solution

Since Y and X are distances, and t is time, and the unit for acceleration is m/s², I put two and two together. That was pretty easy, but I looked at the question properly and saw there was more to it than that, for instance, the 'first two equal and consecutive intervals of time t' I used different equations of motion such as s=ut+1/2at and v²=u²+2aS, but I kept getting stuck. Help, please!

Suppose you start at x=0 and t=0 and accelerate uniformly at acceleration a. Then where are you at time t? Where are you at time 2t? Find X and Y from that and put them into (Y-X)/t^2.

 

[BvU]

Hello VF, welcome to PF

Please be meticulous about your equations. What you have to prove is that a = (Y-X)/t².

Your relevant equations should be like$\qquad$ s=ut+1/2at² (you forgot the square).
And you need to explain what these symbols represent.

And (if necessary) a second equation is v(t) = v₀ + at

But Dick (beat me to it) gives you a way to make do without needing this last equation.

 

[VeeFlamesPhys]

@BvU oops sorry. My smartphone keyboard is wonky(seriously), but that's beside the point.
Yeah, I kind of did what @Dick said earlier, but notice that X and Y are different distances, and do not denote one point to another. The question says that an object will cover these two distances in the same time 2t with a uniform acceleration...
Or is it saying so? I don't know.. I think I'm totally missing something. thanks for the tips, anyway! Really appreciate it.

 

[VeeFlamesPhys]

@BvU i'll be very sure to remember that next time. Thanks!

 

[Dick]

@BvU oops sorry. My smartphone keyboard is wonky(seriously), but that's beside the point.
Yeah, I kind of did what @Dick said earlier, but notice that X and Y are different distances, and do not denote one point to another. The question says that an object will cover these two distances in the same time 2t with a uniform acceleration...
Or is it saying so? I don't know.. I think I'm totally missing something. thanks for the tips, anyway! Really appreciate it.

If you tried to do what I suggested and still can't get it, it would help to post your work so we can see where you are going wrong. Maybe not on the wonky smartphone.

 

[BvU]

My interpretation of

covers distances X and Y in the first two equal and consecutive intervals of time t

• It starts at t=0 from rest at x = 0
• at t = t₁ it is at x = X
• in the interval t from t = t₁ to t₂ = 2t₁ it covers the distance from X to Y

but notice that X and Y are different distances, and do not denote one point to another. The question says that an object will cover these two distances in the same time 2t with a uniform acceleration...

doesn't seem right. X and Y are different, yes. The remainder is unclear or incorrect.

 

[Dick]

My interpretation of

• It starts at t=0 from rest at x = 0
• at t = t₁ it is at x = X
• in the interval t from t = t₁ to t₂ = 2t₁ it covers the distance from X to Y

doesn't seem right. X and Y are different, yes. The remainder is unclear or incorrect.

Agreed, but it doesn't have to start from rest for the formula to be valid.

 

[BvU]

It's justified Dick corrects me. The exercise statement "in the first two equal and consecutive intervals of time t" lured me into interpreting as starting from v = 0. But a = (Y-X)/t² also is true if the particle starts with v = v₀.

Now all VF has to do is come up with equations that feature these distances as functions of time and juggle with the given information.
(the "relevant equation" in post #1 is not enough to solve this. There's nothing with the dimension length (distance) in there. You need more, e.g. from here)

 

[VeeFlamesPhys]

And... SOLVED. Wow. That question really confused me... Thanks for all your help, @BvU and @Dick! :)

 

[VeeFlamesPhys]

My former physics teacher gave me a big hint I totally evaded.