a go-cart moves along a circular track of radius 100ft such that its speed for a short period of time , 0 is less than t which is less than 4s, is v=60(1-e^t^2)ft/s. Determine the magnitude of its acceleration when t=2s?
How do you figure this out while using Simpson's rule with n=50 to evaluate the integral?
To determine the magnitude of the acceleration when t=2s, we need to find the derivative of the velocity function and then substitute t=2 into it.
First, let's find the derivative of the velocity function v(t). The derivative of a function can be found by taking the derivative of each term separately using the power rule and the chain rule.
v(t) = 60(1 - e^(t^2)) ft/s
dv/dt = 60 * (-2t) * e^(t^2) ft/s
= -120t * e^(t^2) ft/s
Now, we'll substitute t=2 into the derivative to calculate the acceleration at t=2s.
dv/dt = -120t * e^(t^2) ft/s
= -120(2) * e^(2^2) ft/s
= -240 * e^4 ft/s
Now that we have the acceleration, we can proceed with using Simpson's rule and n=50 to evaluate the integral of acceleration over the interval from t=0s to t=4s.
Simpson's rule is a numerical method for approximating integrals. It divides the interval into smaller subintervals and uses quadratic approximations to estimate the area under the curve.
To use Simpson's rule, we need to evaluate the integral of acceleration from t=0s to t=4s:
∫[0,4] [ -240 * e^4 ] dt
Now, let's use Simpson's rule with n=50 to evaluate this integral.
1. Determine the step size (h) by dividing the interval (4 - 0) by the number of subintervals (n):
h = (4 - 0) / 50 = 0.08
2. Set up the weights for Simpson's rule:
The weights follow a pattern of 4, 2, 4, 2, ... , 4, 2, 4, where the first and last weights are 4, and the rest alternate between 4 and 2.
3. Calculate the values of the function to be integrated at the nodes:
We need to evaluate -240 * e^4 for each node value.
4. Apply Simpson's rule formula:
∫[0,4] [ -240 * e^4 ] dt ≈ (h/3) * [ 4y0 + 2(y1 + y2 + y3 + ... + y48 + y49) + 4y50 ]
Here, y0 represents the value of -240 * e^4 at t=0, y1 represents the value at t=h, y2 represents the value at t=2h, and so on.
We will substitute these values into the formula and calculate the approximate integral.
5. Substitute the values and calculate:
∫[0,4] [ -240 * e^4 ] dt ≈ (0.08/3) * [ 4(-240 * e^4) + 2(y1 + y2 + y3 + ... + y48 + y49) + 4(-240 * e^4) ]
Evaluating this expression will give you the approximate value of the integral.
Please note that the calculation steps to evaluate the integral using Simpson's rule may vary depending on the software or tool you are using to perform the calculations.