Find the tangential and normal components of the acceleration vector. (r)= (t^2 +1)I +t^3 j, t>=0?
I need to know what are the formulas to solve for them if I got stuck, I will ask later.
acT(t)=(r'(t) dot r''(t))/(llr'(t)ll)
acN(t)=(llr'(t)crossr''(t)ll)/(llr'(t)ll)
To find the tangential and normal components of the acceleration vector, we can follow the steps below:
Step 1: Differentiate the position vector, r(t), twice with respect to time, t, to find the acceleration vector, a(t).
Given:
r(t) = (t^2 + 1)i + t^3j
Differentiate r(t) once:
v(t) = d r(t)/dt = 2ti + 3t^2j
Differentiate r(t) again to find the acceleration:
a(t) = d v(t)/dt = d^2 r(t)/dt^2 = 2i + 6tj
Now we have the vector expression for the acceleration, a(t) = 2i + 6tj.
Step 2: Find the unit tangent vector, T(t), by dividing the velocity vector by its magnitude.
The unit tangent vector, T(t), is given by:
T(t) = v(t)/|v(t)|
First, find the magnitude of the velocity vector, |v(t)|:
|v(t)| = sqrt((2t)^2 + (3t^2)^2) = sqrt(4t^2 + 9t^4)
Then, divide the velocity vector, v(t), by |v(t)|:
T(t) = v(t)/|v(t)| = (2ti + 3t^2j) / sqrt(4t^2 + 9t^4)
Step 3: Find the tangential component of the acceleration, a_t, by projecting the acceleration vector onto the unit tangent vector.
The tangential component of the acceleration, a_t, is given by:
a_t = dot_product(a(t), T(t))
To calculate the dot product, multiply the corresponding components of a(t) and T(t), and then sum them:
a_t = (2i + 6tj) · (2ti + 3t^2j)
= (2 * 2t) + (6t * 3t^2)
= 4t + 18t^3
Therefore, the tangential component of the acceleration is a_t = 4t + 18t^3.
Step 4: Find the normal component of the acceleration, a_n, by subtracting the tangential component from the acceleration vector.
The normal component of the acceleration, a_n, is given by:
a_n = a(t) - a_t
Substitute the values of a(t) and a_t:
a_n = (2i + 6tj) - (4t + 18t^3)
= 2i + 6tj - 4t - 18t^3
Therefore, the normal component of the acceleration is a_n = 2i + 6tj - 4t - 18t^3.
These are the tangential and normal components of the acceleration vector. If you have any more questions or any specific step you need further explanation on, feel free to ask.
well, it's been a while since I did those, and it's late.
I was going to consult google first.
If all you want is the formulas, I'm sure you'll find them there.
Or check out differential geometry. Those ideas are among the first presented.
I'll check back tomorrow...