Average Velocity vs. Instantaneous Velocity

In summary, the conversation discusses the relationship between average velocity and instantaneous velocity for an object undergoing uniformly accelerated motion. It is explained that average velocity is the slope of the velocity over a certain interval, and instantaneous velocity is the slope at a certain point on the interval. The conversation also mentions that the acceleration graph is the graph of the slope of the velocity function, and that the slope of this graph is linear. It is stated that the average velocity and instantaneous velocity are equal at the midpoint between two given times.
  • #1
crazyecto
1
0
The graph below shows a plot of velocity vs. time for an object undergoing uniformly accelerated motion. The object has instantaneous velocity v1 at time t1 and instantaneous velocity v2 at time t2. Use the graph and the fact that, when the acceleration is constant, the average velocity can be written as vavg = (v1 + v2)/2 to explain how we know that the instantaneous velocity at the midpoint time is equal to vavg on the interval.

V = Vo + at

X - Xo = Vot + .5at2

v2 = vo2 + 2a(X - Xo)

X - Xo = .5(Vo + V)t

Now I know the answer has to do with some how deriving one of these formulas to get it to equal the formula for average veolcity but I can't seem to remember how. I think you some how take the derivative of X - Xo = .5(Vo + V)t to get the answer maybe?
 
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  • #2
What graph?
 
  • #3
pretty sure I know what the graph looks like cause I just had this problem in a homework lol.

This answer doesn't require any deriving. Average velocity is simply the slope of the velocity over a certain interval, and instantaneous velocity is the slope at a certain point on the interval. We know that the acceleration graph is simply the graph of the slope of the velocity function.The slope of this graph is clearly linear. And because average velocity and instantaneous velocity are just values of the slope over their intervals or at certain points, if we take the average velocity from t1 to t2 it would be the same as the velocity at the midpoint between t1 and t2.

Hopefully that makes some sense.
 
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  • #4
Stryder_SW said:
pretty sure I know what the graph looks like cause I just had this problem in a homework lol.

This answer doesn't require any deriving. Average velocity is simply the slope of the velocity over a certain interval, and instantaneous velocity is the slope at a certain point on the interval. We know that the acceleration graph is simply the graph of the slope of the velocity function.The slope of this graph is clearly linear. And because average velocity and instantaneous velocity are just values of the slope over their intervals or at certain points, if we take the average velocity from t1 to t2 it would be the same as the velocity at the midpoint between t1 and t2.

Hopefully that makes some sense.

you do realize that s/he posted this over a year ago, and probably doesn't need it anymore
 
  • #5
mmmm no unfortunately I did not realize that...this was a colossal waste of time.
 

Related to Average Velocity vs. Instantaneous Velocity

1. What is the difference between average velocity and instantaneous velocity?

The average velocity of an object is the average rate at which it changes its position over a specific time interval. It is calculated by dividing the change in position (displacement) by the change in time. On the other hand, instantaneous velocity is the velocity of an object at a specific moment in time. It is calculated by taking the derivative of the position function with respect to time.

2. How are average velocity and instantaneous velocity represented mathematically?

Average velocity is represented as vavg = Δx/Δt, where Δx is the change in position and Δt is the change in time. Instantaneous velocity is represented as v = dx/dt, where dx is the infinitesimal change in position and dt is the infinitesimal change in time.

3. Can an object have the same average velocity and instantaneous velocity?

Yes, an object can have the same average velocity and instantaneous velocity if its motion is uniform (constant velocity). In this case, the object's velocity does not change over time, so both average and instantaneous velocity will have the same value.

4. How do average velocity and instantaneous velocity relate to an object's position-time graph?

The average velocity of an object is equal to the slope of the line connecting two points on the position-time graph. On the other hand, the instantaneous velocity at a specific point on the graph is equal to the slope of the tangent line at that point.

5. What are the units for average velocity and instantaneous velocity?

The units for average velocity are distance per time, such as meters per second (m/s) or kilometers per hour (km/h). The units for instantaneous velocity are also distance per time, but because it represents velocity at a specific moment, it is often written as m/s at t=0 to indicate the time at which it is measured.

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