Equations of Motion (Deriving equations)

[arjun90]

Hi, I am just wondering how you would approach this problem:
Using the definitions below, derive an equation for velocity as a function of acceleration and time (v=f(a,t)). Assume initial velocity is Vo. The answer to this problem is v=v0+a(t2-t1). My question is how would you arrive to this answer step-by-step. Below are the definitions:

x=current position in the x dimension
deltax= change in position
t=time now, t0 is the starting time.
deltat= a time interval, t2-t1.
v=deltax/deltat (use as a scaler for now).
deltav= a change in velocity.
a=deltav/deltat (Use as a scaler for now).

Subscripts: 0 is an initial value, other numbers are subsequent values in time order as needed.

v (average)= (v1+v2)/2, a simple average.

Any help will be appreciated. Thank you.

 

[sokratesla]

# I would first look at the definitaion of a
$$a(t) = \frac{dv(t)}{dt}$$
# Integrate both sides
$$\int^{t}_{t_0} a(t) dt = v(t) - v(0)$$
# This is the relation between v and a. but it is not in the form of you want, i.e. $$v=f(a,t)$$. It is an integral equation.
# So, there should be an assumption about a, which changes integral the relation between a and v to a function. I think if you think, you can find it by yourself.

 

[ogb p]

To derive an equation for velocity as a function of acceleration and time, we can start by using the definition of average velocity: v (average) = (v1 + v2)/2. We can also use the definition of acceleration: a = deltav/deltat.

Next, we can substitute in the values for v1 and v2 using the definitions provided: v1 = v0 + deltax0/deltat and v2 = v0 + deltax/deltat.

Substituting these values into the equation for average velocity, we get: v (average) = (v0 + deltax0/deltat + v0 + deltax/deltat)/2.

Simplifying this, we get: v (average) = (2v0 + deltax0 + deltax)/2deltat.

We can also use the definition of velocity as deltax/deltat to rewrite this as: v (average) = (2v0 + v*deltat)/2deltat.

Now, we can rearrange this equation to solve for v: v = (2v0 + v*deltat)/2deltat.

Multiplying both sides by 2deltat, we get: 2deltat*v = 2v0 + v*deltat.

Subtracting v*deltat from both sides, we get: 2deltat*v - v*deltat = 2v0.

Factoring out v, we get: v*(2deltat - deltat) = 2v0.

Simplifying, we get: v*deltat = 2v0.

Finally, solving for v, we get: v = 2v0/deltat.

This is the equation for velocity as a function of acceleration and time. We can also rewrite this as v = v0 + a(t2-t1), which is the same answer provided in the problem.

I hope this step-by-step explanation helps you understand how to arrive at the answer. Please let me know if you have any other questions.