- Since the speed is being reduced at a "uniform rate", the rate of change of speed v_dot, is a constant.
- You will need to use the chain rule of differentiation followed by an integration over the path in order to determine v_dot (you are given initial and final speeds at points A and C, respectively, that will be used in this integration).
- For a) (immediately before B), the path is straight, and therefore has only a tangential component of acceleration (and friction).
- For b) (immediately after B), the path is circular with BOTH tangential and normal components of acceleration (as well as tangential and normal components of friction).
- For c), (immediately before C), the speed is zero and with a non-zero v_dot. Therefore, the normal component of acceleration (and friction) is zero.
Sunday, September 30, 2007
Homework Hint: Problem 3/91
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7 comments:
When we use the chain rule of differentiation should our formula be: v(dv/dt)? Then we take the integration of v(dv) = a(dt)? Or is that totally wrong?
v = dx/dt and a = dv/dt. Solving for dt and separating the variables you get V dv = a dx. Then, integrate both sides respectively. You were half way there.
i didn't use chain rule of differentiation to find answer for this problem. you know the total distance to C, and you know that it will stop at C. so you can use this equation: v^2 = (v_o)^2 + 2a(x-x_o). you find deceleration from this equation (and it is a_tan). So you use that to find part a and c. for b, you have to find a_nor. we know that a_n = v^2 / p. Be careful that you have to find v at point b. use the same equation above using a_tan as a. After finding a_nor. you can simply find a = sqrt(a_tan^2 + a_nor^2). and you can use that for part b.
Even if you think you didn't use it, you still did. You integrated V dv = a dx (which you obtained via the chain rule) which you then expanded into v^2 = (v_o)^2 + 2a(x-x_o).
You can be clever and apply some energy conservation equations
work = force * distance
ke = 1/2 * m * v^2
delta(work) = delta(ke)
ah i guess i did use it then, it is useful to memorize those things
All good thoughts.
I like andrew crandall's the best: both the constant acceleration equation and the work-energy equation come from the chain rule and integration. The fewer things we need to memorize, the fewer things we will forget.
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