How Can Linear Motion Equations Explain Particle Dynamics and Car Acceleration?

[mcintyre_ie]

Hey
Ok, so I am having troule with yet another accelerated linear motion question...

any help is desperately needed and very much appreciated :)

So here it goes:

(A) A particle is projected vertically upwards with velocityum/s and is at a height h after t1 and t2 seconds respectively. Prove that:
t1.t2 = (2h)/g

(B) A car accelerates uniformly from rest to a speed v m/s. It then continues at this constant speed for t seconds and then decelerates uniformly to rest.
The average speed for the journey is (3v)/4

(i) Draw a speed-time graph and hence, or other wise, prove that the time for the journey is 2t seconds.

(ii) If the car driver had observed the speed limit of (1/2)v, find the least time the journey would have taken, assuming the same acceleration and deceleration as in (i).

Ok, so for part a, I've been told that "setting those equal should make it easy to get the result". Basically what i took that up as meaning is " h=h " so "(-g/2)t1^2+ u t1 = (-g/2)t2^2+ u t2 " , which gives me two variables t1 and t2, equating to each other. Where am i going wrong here? Is there some thing about the times that I am missing or maybe something else pretty obvious that I am missing? When i try to "fix" my answer i get " (2h)/g = 8ut - 4t^2 ". Which is very very wrong...

So that was part a, now onto part b:

Ive drawn the graph, and made a very dodgy comp reproduction:

Graph Pic

All i can make out is that the area of space 1 is .5(t1)(v) = distance, area of space 2 is (t)(V) and area of space 3 is (.5)(t2)(v).
So again I've got 3 variables (t1, t2 and V). V i can keep, since the average speed is given as being = (3v)/4 (i think this is right anyway). So now that I've done some very basic (very possible that its just basically wrong too...) stuff, can anybody give me a little more help ASAP?
Thanks in advance

 

[HallsofIvy]

I suggested that, since h= (-g/2)t₁²+ u t₁ and
h= (-g/2)t₂²+ u t₂ and you are asked about when the two have
the same height, you try setting them equal:

h=(-g/2)t₁²+ u t₁(-g/2)t₂²+ u
t₂.

Now, Combine those! (-g/2)(t₁²-t₂²)+
u(t₁-t₂)= 0.


Major hint! FACTOR t₁²-t₂².

"All i can make out is that the area of space 1 is .5(t1)(v) = distance, area of space 2 is (t)(V) and
area of space 3 is (.5)(t2)(v). So again I've got 3 variables (t1, t2 and V)."

Actually, your graph is very good. You have a trapezoid with height v. The top "base" is t and
the bottom base is t₁+ t+ t₂ so the area is (1/2)v(t₁+ 2t+
t₂). The average speed is that area divided by the total time,
t₁+ t+ t₂ that gives
(1/2)v(t₁+ 2t+ t₂)/(t₁+ t+ t₂)= 3v/4 or
(t₁+ 2t+ t₂)/(t₁+ t+ t₂)= 3/2.
(t₁+ t₂+ 2t)= (3/2)(t₁+t₂) + (3/2)t so
(1/2)(t₁+ t₂)= -(1/2) t. That is, t₁+ t₂= t. The total
time the vehicle is moving is (t₁+ t₂)+ t= 2t !

 

[mcintyre_ie]

Part A:

Ok, so I am trying to prove that t1.t2 = (2h)/g , and I've got to try and get that from "h=(-g/2)t12+ u t1(-g/2)t22+ ut2" - even with your "major hint" (?!?) I am a little lost still. My brain has gone into serious meltdown over the summer vacation... I've got a very very vague idea of how it *might* come out, but i need another "major" hint at least.

Part B:
(i)
We don't actually have trapezoids on our curriculum, so i was trying to take the area in 3 segments, as you can see from my diagram, which added *slightly* to the confusion.
Towards the end of the proof you say

(1/2)(t1+ t2)= -(1/2) t. That is, t1+ t2= t

Im a little confused as to where the minus sign went after you multiplied across by two. Is it just because a minus answer wouldn't make any sense or is there something else I am not getting?

(ii)
If the car driver had observed the speed limit of (1/2)v, find the least time the journey would have taken, assuming the same acceleration and deceleration as in (i).

Regarding this one, I am a little unsure as how to start off, i know the the maximum speed is now (1/2)v, instead of v in the last question... Thats about as much as i know as being true, regading the rest, I am pretty lost, help! please!

Thanks again for the help and send some more my way asap!

 

[HallsofIvy]

(A) A particle is projected vertically upwards with velocityum/s and is at a height h after t1 and t2 seconds respectively. Prove that:
t1.t2 = (2h)/g

t1 and t2 are both solutions to the quadratic equation

(-g/2)t^2+ vt= h which is the same as (g/2)t^2- vt+ h= 0
which is again the same as t^2- (2v/g)t+ 2h/g= 0

Saying that t1 and t2 satisfy t^2- (2v/g)t+ 2h/g= 0 means that

(t-t1)(t-t2)= t^2- (2v/g)t+ 2h/g.

Multiply that out and see what happens.