# Simple physics question

willworkforfood
Is it true that if the rate of decceleration is two times slower than the rate of acceleration for the same object, that it will need two times the distance to come to a stop as it did to accelerate?

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## Answers and Replies

Mentor
The answer to this question depends on if this is a homework question...

willworkforfood
It is but it is for extra credit :rofl:

Gold Member
It's not a very dificult question. Find the distance traveled under constant acceleration for a given period of time. Use what you know about how much the velocity change to solve for the time taken in terms of the maximum velocity and acceleration. Plug this into the equation you had for the distance travelled. Now you have the distance traveled in terms of the acceleration and maximum velocity. By seeing how the acceleration varies with distance your problem is solved.

Mentor
Figure it out for yourself! How does the distance required to accelerate from a speed of 0 to V (or decelerate from V to 0) depend on the acceleration? (Work out or look up the kinematic formulas for uniformly accelerated motion.)

willworkforfood
Doc Al said:
Figure it out for yourself! How does the distance required to accelerate from a speed of 0 to V (or decelerate from V to 0) depend on the acceleration?

I have no idea where to start or how the distance is related to the acceleration.

Mentor
Are you familiar with the kinematics of uniformly accelerated motion? (If not, then you are not ready to solve this problem.)

willworkforfood
I know that the equation for acceleration is final velocity - initial velocity divided by time, but I really don't see how that helps answer my original question :(

Mentor
willworkforfood said:
I know that the equation for acceleration is final velocity - initial velocity divided by time
That's a start: You have a relation connecting acceleration and velocity (and time). Now figure out how to get the distance. (Hint: Distance equals average speed X time.)

willworkforfood
The problem doesn't give me average speed or time though.

Mentor
willworkforfood said:
The problem doesn't give me average speed or time though.
You don't need actual values. You are trying to find the relationship between acceleration and distance. So far you have:
(1) a = v/t (the initial speed is zero)

I suggest you combine that with my "hint" about distance and average speed. (If something goes from 0 to V, what is its average speed?) Write my statement mathematically. Then you'll have two equations that you can combine to relate distance with acceleration.

willworkforfood
So equation 2 would be x = v*t and you substitute v = at and get

x = a*t^2

Is that right?

Wow I feel stupid thanks for the help on that.

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Mentor
willworkforfood said:
So equation 2 would be x = v*t
No.... you need average speed, which is v/2, not v.
and you substitute v = at and get

x = a*t^2
When you combine the equations, don't eliminate v; instead, eliminate t. You want to end up with an equation relating v, a, and x.

willworkforfood
So instead equation 2 would be

x=(vt)/2

Solving for t would be

t = 2x/v

Substituting that into equation 2 would be:

a= v * (v/2x) or a = v^2 / 2x

Meaning when acceleration is 2 times decceleration then you need half the distance to accelerate? Thanks I think I get it now, I am not worthy :P

Mentor
Exactly. The derived relationship a = v^2 / 2x tells you that acceleration and stopping distance are inversely related (for a given maximum speed).

Note that this same equation can help you understand why, for a given deceleration, that if you are moving twice as fast it takes 4 times the distance to stop.