Derivations of the motion equations

In summary, the conversation is about practicing the derivations of the Motion Equations, specifically the last one, and the difficulty in getting the fractions to add and subtract correctly. The individual is seeking help with identifying the mistake and being reminded not to skip steps in order to avoid making mistakes with signs.
  • #1
mateomy
307
0
Practicing the derivations of the Motion Equations, and I am looking through my notes that I transcribed from my professor...Im getting caught up on one spot.

Im getting all the equations to "pop out" except for this last one, which, I can see; but, the algebra is throwing me off...

Finishing off with these as the fundamentals to get the last derivation:

[tex]
x=v_0 + at ; t= \frac{v-v_0}{a} ; x=x_0 + v_0 t + \frac{1}{2}at^2
[/tex]

Substituting the new t value into the latter equation and expanding...

[tex]
x= x_0 + \frac{v_0 v}{a} - \frac{v_0^2}{a} + \frac{v^2}{2a} - \frac{v_0^2}{2a} - \frac{v_0 v}{a}
[/tex]

I can't seem to get the fractions to add and subtract out in the right way so that I can get the final equation of

[tex]
x= x_0 + \frac{1}{2a}(v^2 - v_0^2)
[/tex]

Specifically I can't get the -v(initial)^2/a and the -v(initial)^2/2a to add up so that I can factor (in the final equation) out the 1/2. I know I am doing something absent minded. Can somebody please point it out? Thank you in advance for any pointers.
 
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  • #2
try not to skip steps =P it makes it really easy to make mistakes with signs
 
  • #3
mateomy said:
[tex]
x= x_0 + \frac{v_0 v}{a} - \frac{v_0^2}{a} + \frac{v^2}{2a} - \frac{v_0^2}{2a} - \frac{v_0 v}{a}
[/tex]
Looks like you messed up the expansion of the t2 term. (You have an extra minus sign on one of the terms.)
 
  • #4
Doc Al said:
Looks like you messed up the expansion of the t2 term. (You have an extra minus sign on one of the terms.)

Okay, I'll recheck that expansion. Thanks!
 
  • #5


It seems like you may have made a small mistake in your algebra. Let's go through the steps together to see if we can figure out where the issue is.

First, let's start with the equation you are trying to derive:

x=x_0 + \frac{1}{2}at^2

We know that t= \frac{v-v_0}{a} from one of the given equations. So let's substitute that into our original equation:

x=x_0 + \frac{1}{2}a(\frac{v-v_0}{a})^2

Now, let's simplify the right side of the equation:

x=x_0 + \frac{1}{2}a(\frac{v^2-2vv_0+v_0^2}{a^2})

Next, we can distribute the a to the terms inside the parentheses:

x=x_0 + \frac{1}{2}(v^2-2vv_0+v_0^2)

Now, let's combine like terms:

x=x_0 + \frac{1}{2}(v^2-2vv_0+v_0^2)

x=x_0 + \frac{1}{2}(v^2-2vv_0+2v_0^2-v_0^2)

x=x_0 + \frac{1}{2}(v^2-2vv_0+2v_0^2)-\frac{1}{2}v_0^2

x=x_0 + \frac{1}{2}(v^2-2vv_0+2v_0^2)-\frac{1}{2}v_0^2

x=x_0 + \frac{1}{2}(v^2-2vv_0+v_0^2)-\frac{1}{2}v_0^2

x=x_0 + \frac{1}{2}(v-v_0)^2-\frac{1}{2}v_0^2

Finally, we can factor out the 1/2 from the first two terms:

x=x_0 + \frac{1}{2}(v-v_0)^2-\frac{1}{2}v_0^2

x=x_0 + \frac{1}{2}(v-v_0)^2-\frac{1}{2}
 

1. What are the basic motion equations?

The basic motion equations are the equations that describe the relationship between an object's position, velocity, and acceleration. These equations are:

Position (x) = Initial position (x0) + Initial velocity (v0) * Time (t) + 0.5 * Acceleration (a) * Time (t)2

Velocity (v) = Initial velocity (v0) + Acceleration (a) * Time (t)

Acceleration (a) = Change in velocity (v - v0) / Time (t)

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

Average velocity is the total displacement of an object divided by the total time it took to cover that displacement. Instantaneous velocity, on the other hand, is the velocity of an object at a specific moment in time. It is calculated by taking the limit of the average velocity as the time interval approaches zero.

3. How do you derive the motion equations?

The motion equations can be derived using the principles of calculus. The first equation, position as a function of time, can be derived by integrating the velocity equation with respect to time. The second equation, velocity as a function of time, can be derived by taking the derivative of the position equation with respect to time. The third equation, acceleration as a function of time, can be derived by taking the derivative of the velocity equation with respect to time.

4. What is the significance of the motion equations?

The motion equations are significant because they allow us to mathematically describe the motion of objects. By using these equations, we can determine an object's position, velocity, and acceleration at any given time. They are used in many fields of science and engineering, such as physics, astronomy, and aerospace engineering.

5. Are there any limitations to the motion equations?

Yes, there are some limitations to the motion equations. They are based on the assumptions that the object is moving along a straight line with constant acceleration and there are no external forces acting on the object. In reality, most objects do not move in a straight line and are affected by various forces, so the equations may not accurately describe their motion. Additionally, the equations only work for objects moving at a constant acceleration, and cannot be applied to objects with changing accelerations.

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