Exploring a(t), v(t), and x(t) Relationships

In summary, the conversation is discussing the equations for acceleration (a), velocity (v), and position (x) given a function for displacement (x). The correct equations are v(t) = (alpha)t^2/2 + w and x(t) = ((alpha)t^2)/6 + wt + a. The discrepancy in the answer is due to the incorrect use of the constant k, which should be evaluated using the given time and displacement values. The correct answer is ((alpha)t^2)/6 + wt + a - 2w - 4(alpha)/3.
  • #1
xpack
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http://img40.imageshack.us/img40/8591/ques.png [/URL]



a(t)=dv/dt
v(t)=dx/dt



I thought it was just
v(t)= (alpha)t^2/2 + w
x(t)=((alpha)t^2)/6 + wt + a

But the answer is ((alpha)t^2)/6 + wt + a - 2w - 4(alpha)/3

Can someone please explain this to me...
 
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  • #2
Starting from your v = .5*@*t^2/2 + W
x is the integral dt of this: x = .5*@/3*t^3 + wt + k
To evaluate the constant k, put in time 2 and x = A:
This gives you k = A - 4/3@ - 2w
Put that back in the x formula and you have the answer you are looking for, though there seems to be a t^2 instead of a t^3 in the answer.
 
  • #3
Thank you very much!
 

FAQ: Exploring a(t), v(t), and x(t) Relationships

What are a(t), v(t), and x(t) relationships?

A(t), v(t), and x(t) relationships refer to the mathematical relationships between acceleration (a), velocity (v), and displacement (x) over time (t). These relationships are often studied in physics and other scientific fields to understand the motion of objects.

How are a(t), v(t), and x(t) related?

Acceleration (a) is the rate of change of velocity (v), and velocity (v) is the rate of change of displacement (x). This means that acceleration is the second derivative of displacement with respect to time (a = d2x/dt2), and velocity is the first derivative of displacement with respect to time (v = dx/dt).

How can a(t), v(t), and x(t) be graphed?

A(t), v(t), and x(t) can be graphed on a coordinate plane with time (t) on the x-axis and the respective values (a, v, or x) on the y-axis. The resulting graphs will show how the values change over time and can provide information about the motion of an object.

What is the significance of the slope of a(t) and v(t) graphs?

The slope of an a(t) graph represents the rate of change of acceleration over time, while the slope of a v(t) graph represents the rate of change of velocity over time. These slopes can provide information about the acceleration or deceleration of an object.

How are a(t), v(t), and x(t) used in real-world applications?

A(t), v(t), and x(t) relationships are used in a variety of real-world applications, such as studying the motion of objects in physics, analyzing data from sports and fitness trackers, and designing transportation systems. They can also be used to make predictions about future motion and to optimize performance in various industries.

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