An electron, starting from rest and moving with a constant acceleration, travels 2.0 cm in 5.0 ms. What is the magnitude of this acceleration?

- anonymous

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- JamesJ

What's the equation of motion for a body under constant acceleration?

- JamesJ

Suppose
v = initial velocity
t = time
a = constant acceleration
Then what is the formula for the distance, d, the object moves in time t?
d = ... what?

- Kainui

What is the definition of acceleration? It's the change of velocity with respect to time. How would you put this in the form of an equation?

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- JamesJ

I'm going to change notation slightly, writing now u for initial velocity. Then we have:
u = initial velocity
t = time
a = constant acceleration
Write also
v(t) = velocity of object at time t
d(t) = position of object at time t
Now, by definition,
acceleration = (change in velocity)/(Time)
hence
\[ a = \frac{v(t) - u}{t} \]
thus
\[ v(t) - u = at \]
and
\[ v(t) = u + at \]
Making sense so far?

- anonymous

Yeah, I'm following

- JamesJ

Now, what's d(t) ?

- anonymous

The position?

- JamesJ

yes, what's the formula for it?

- anonymous

v1t + 1/2 a t^2?

- JamesJ

Yes
\[ d(t) = ut + \frac{1}{2}at^2 \]
For your problem, you're told that the initial velocity is zero, t = 0.5 ms and
d(0.5 ms) = 0.92 m
Now calculate a.

- JamesJ

*correction, d(0.5 ms) = 0.02 m, not 0.92.

- anonymous

would I try and get a by its self?

- JamesJ

Thus
u = 0 m/s
t = 0.005 s
d = 0.02 m
Substitute that into the equation above and solve for a

- anonymous

.02 = 0(.005) + .5(a)(.005)^2?

- JamesJ

yes

- anonymous

So if a=(v(t)−u)/t, that would equal v(t)/t, correct?

- JamesJ

yes

- JamesJ

but here velocity is not constant so you have to be very careful with that formula

- anonymous

But the velocity equation is dependent on the acceleration equation: v(t)=u+at

- JamesJ

Yes, so here because acceleration is constant, we're good. I.e., the average acceleration is equal to the constant acceleration.

- anonymous

But the closest we got to solving for a is: .02 = 0(.005) + .5(a)(.005)^2
and v(t) needs a,
v(t)=u+at
I'm not getting this part, how do we solve for either one?

- anonymous

?

- JamesJ

The question only asks you to find the acceleration. Hence you only need one equation and the one that is germane here--because it uses the information of the problem--is the the one into which you've already substituted.

- JamesJ

Hence you have:
.02 = 0(.005) + .5(a)(.005)^2
This already simplifies to
\[ \frac{0.005^2}{2}a = 0.02 \]
Now solve for \( a \).

- anonymous

A = .02(2/.005^2) = .04/.005^2?

- JamesJ

Evaluate it. You can't leave the answer in that form.

- anonymous

1600?

- anonymous

What about the units though, is that in meters?

- JamesJ

We were (or at least I was) careful to convert all the units into SI units before we started calculating
time = seconds
distance = meters
velocity = meters/sec = m/s
acceleration = m/s^2

- anonymous

Ah, so its 1.6 km/s^2?

- JamesJ

yes

- anonymous

Awesome! Thanks for all your help!

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