A question on kinematics

• amitchhajer
In summary, the equation \frac{\sqrt{2u^2+2v^2}}{2} represents the velocity of the midpoint of a train with length s, acceleration a, front part passing a point with velocity v, and end part passing the same point with velocity u. This can be proven by using kinematic equations and graphically representing the train's speed as a vector on the XY plane. The logic used to derive this equation is based on the assumption that all points on the train are connected, therefore traveling at the same velocity.

amitchhajer

A train having some lenght.its front part passes through a point 'N' with velocity 'v' while its end part passes through same point with velocity 'u'.Prove that the mid point passes through the same point with velocity √v2+u2 /2.all the parts has acceleration a.

amitchhajer said:
Prove that the mid point passes through the same point with velocity √v2+u2 /2.
I'm sorry, is this the equation you're trying to prove?

$$\frac {\sqrt{v^2+u^2}}{2}$$

Or this?

$$\frac {\sqrt{2v+2u}}{2}$$

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Let the length of the train be s, then midpoint travelles a distance s/2 after the front passes the point. Try to set equations for s and s/2

amitchhajer said:
A train having some lenght.its front part passes through a point 'N' with velocity 'v' while its end part passes through same point with velocity 'u'.Prove that the mid point passes through the same point with velocity √v2+u2 /2.all the parts has acceleration a.

It's been a couple of days since this was posted, and I started obsessing on it! Here's how I did it. (Warning! I do not get the solution shown and the way I did it seems much to difficult for k-12 problem!)

Let L be the length of the train and T the time interval between the front of the train passing point "N" until the end of the train passes point "N". Assuming that acceleration a is a constant, $a= \frac{u- v}{T}$.

The basic kinematic equation are now $v(t)= v+ \frac{u-v}{T}t$
and $L(t)= vt+ \frac{u-v}{2T}t^2$ where v(t) is the velocity of the train at time t after the front passes point "N" and L(t) is the distance the front of the train has gone in time t.

Since, by definition of L and T, the front of the train will have gone distance L in time T, we have
$$vT+ \frac{u-v}{2T}T^2= vT+ \frac{u-v}{2}T= L$$
$$/frac{u+v}{2}T= l$$ so
$$T= \frac{2L}{u+v}$$
Putting that value for T in the two equations
$$v(t)= v+ \frac{u^2- v^2}{2L}t$$ and
$$L(t)= vt+ \frac{u^2- v^2}{4L}t^2$$.
We can use that L(t) equation to determine the time when the middle of the train passes point "N":
$$L(t)= vt+ \frac{u^2- v^2}{4L}t^2= L/2$$ or
$$t^2+ \frac{4Lv}{u^2- v^2}t= \frac{2L^2}{u^2- v^2}$$
Completing the square:
$$t^2+ \frac{4Lv}{u^2- v^2}t+ \frac{4L^2v^2}{(u^2-v^2)^2}= \frac{2L^2(u^2-v^2)+ 4L^2v^2}{(u^2-v^2)^2}$$
$$(t+ \frac{2Lv}{u^2-v^2})^2=\pm\frac{L\sqrt{2(u^2+v^2}}{u^2-v^2}$$
$$t= \frac{L\sqrt{2(u^2+v^2)}-2v}{u^2-v^2}$$

Now plug that into $$v(t)= v+ \frac{u^2-v^2}{2L}t$$ (noting that both the "L" and "u2-v2" terms cancel) we get, for the speed at the time the middle of the train passes point "N":
$$\frac{\sqrt{2(u^2+v^2)}}{2}- v$$.

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Alternative solution

All points on the train are connected, so travel at the same velocity.
i.e. the front of the train is traveling at the same velocity as the end of the train, so when the end of the train reaches velocity 'u', so also is 'u' the velocity of the front of the train. And by this tine the train has traveled a distance L
When the mid-point of the train reaches 'N', the front of the train will have traveled a distance ½L.

Let a be the (constant) accln of the train.

u² = v² + 2as
or,
u² = v² + 2aL
a = (u² - v²)/(2L)
==============

To find velocity, w say, of mid-point of train when reaching 'N', i.e. after having traveled ½L

w² = v² + 2a½L
w² = v² + aL
w² = v² + (u² - v²)/(2L)*L
w² = v² + (u² - v²)/2
w² = (u² + v²)/2
============

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Let the length of the train be L, and the acceleration be a.
The rear end of the train passes the point after the train has traveled distance L, so that
v^2 - u^2 = 2*a*L
The mid point of the train passes the point after the train has teavelled distance L/2, with velocity v' hence
v'^2 - u^2 = 2*a*(L/2)

Am I onto a graphical solution?

On the XY plane, let the horizontal axis be the train's speed when its front passes a mark, at time 0. Let the vertical axis be the speed when its rear passes the same mark at time T. Connect point v on the hor. axis to point u on the ver. axis with a straight line. I have a triangle with hypotenuse length = $\sqrt{u^2+v^2}$.

One can visualize the train's speed moving with uniform acceleration from the horizontal intercept (at t = 0) to the vertical intercept (at t = T) on the hypotenuse. At the midpoint, the speed is the length of the vector that connects the midpoint of the hypothenuse with the origin = $\sqrt{u^2+v^2}\left/2\right.$.

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for some reason I got HallsofIvy's answer, minus the minus v
let $$s =$$ total length of train
let $$t =$$total time from front at point N to back at point N
let $$t_2 =$$ time when midpoint reaches M
$$\frac{u-v}{t} = a$$
$$s = vt + \frac{1}{2}at^2$$
$$\frac{s}{2} = vt_2+ \frac{1}{2}at^2_2$$
$$s = 2\frac{s}{2}$$
$$vt + \frac{1}{2}at^2 = 2vt_2 + at^2_2$$
$$at^2_2 + 2vt_2 - (vt + \frac{1}{2}at^2 )= 0$$
$$t_2 = \frac{-2v \pm\sqrt{4v^2+4at(\frac{1}{2}at+v)}}{2a}$$
I am too lazy to type out the steps where I simplify the radicand
the $$u$$ comes from substituting the acceleration for the formula
$$t_2 = \frac{-2v + \sqrt{2u^2+2v^2}}{2a}$$
let $$v_m =$$ velocity at midpoint when passing M
$$v_m = v + at_2$$
$$v_m = v + a \frac{-2v + \sqrt{2u^2+2v^2}}{2a}$$
$$v_m = v + \frac{-2v + \sqrt{2u^2+2v^2}}{2}$$
$$v_m = \frac{\sqrt{2u^2+2v^2}}{2}$$

how is my logic flawed?

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1. What is kinematics?

Kinematics is the branch of physics that studies the motion of objects without considering the forces that cause the motion.

2. What are the three main concepts in kinematics?

The three main concepts in kinematics are displacement, velocity, and acceleration.

3. What is the difference between speed and velocity?

Speed is a scalar quantity that measures the rate of motion without considering direction, while velocity is a vector quantity that measures the rate of motion and includes direction.

4. How is acceleration calculated?

Acceleration is calculated as the change in velocity over the change in time, expressed as a vector quantity.

5. What is the difference between average and instantaneous acceleration?

Average acceleration is calculated over a specific time interval, while instantaneous acceleration is the acceleration at a specific moment in time.