Question

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?

Answer 1

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

Answer 2

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?

Answer 3

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?

Answer 4

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?

Answer 5

Yeah, I'm following

Answer 6

Now, what's d(t) ?

Answer 7

The position?

Answer 8

yes, what's the formula for it?

Answer 9

v1t + 1/2 a t^2?

Answer 10

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.

Answer 11

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

Answer 12

would I try and get a by its self?

Answer 13

Thus
u = 0 m/s
t = 0.005 s
d = 0.02 m

Substitute that into the equation above and solve for a

Answer 14

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

Answer 15

yes

Answer 16

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

Answer 17

yes

Answer 18

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

Answer 19

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

Answer 20

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

Answer 21

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?

Answer 22

?

Answer 23

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.

Answer 24

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 \( a \).

Answer 25

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

Answer 26

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

Answer 27

1600?

Answer 28

What about the units though, is that in meters?

Answer 29

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

Answer 30

Ah, so its 1.6 km/s^2?

Answer 31

yes

Answer 32

Awesome! Thanks for all your help!