A car travels along the curved path with radius of curvature p= 30m such that its speed is increased by at = (0.5e^t) m/s^2, where t is in seconds. determine the magnitudes of its velocity and acceleration after the car has traveled s = 18m starting from rest. neglect the size of the car.
Above answer is completely wrong.. it seems he forgot that it's an initial value problem
The distance s travelled in time t' is
S (t') = Double integral a_t dt
t=0 to t'
18 = 0.5*e^t' meters
t' = ln 36 = 3.584 s
V (t') = Single integral a_t dt
t=0 to t'
= 0.5*e^t' = 18 m/s
a_t (t') = 18 m/s^2
There is also a centrifugal acceleration component,
a_r = V(t')^2/R = 324/30 = 10.8 m/s^2
To determine the magnitudes of the car's velocity and acceleration after traveling 18m starting from rest, we need to calculate the speed and acceleration at that point.
First, let's find the time it takes for the car to travel 18m using the given acceleration function. We can do this by integrating the acceleration function to get the velocity function and then integrating the velocity function to get the position function.
Given:
Acceleration, a(t) = 0.5e^t m/s^2
Initial velocity, v(0) = 0 m/s
Distance traveled, s = 18 m
1. Find the velocity function:
Integrate the acceleration function with respect to time:
v(t) = ∫(0.5e^t) dt
To integrate e^t, we use the formula for the integral of e^x, which is e^x:
v(t) = 0.5∫(e^t) dt
= 0.5e^t + C
Now we need to find the constant of integration, C. Since the initial velocity is 0 (v(0) = 0), we can substitute this into the velocity function:
v(0) = 0.5e^0 + C
0 = 0.5 + C
C = -0.5
So, the velocity function is:
v(t) = 0.5e^t - 0.5
2. Find the position function:
Integrate the velocity function with respect to time:
s(t) = ∫(0.5e^t - 0.5) dt
To integrate 0.5e^t, we use the formula for the integral of e^x, which is e^x:
s(t) = 0.5∫(e^t) dt - 0.5∫(1) dt
= 0.5e^t - 0.5t + C'
Now we need to find the constant of integration, C'. Since the initial position is 0 (s(0) = 0), we can substitute this into the position function:
s(0) = 0.5e^0 - 0.5(0) + C'
0 = 0.5 - 0 + C'
C' = -0.5
So, the position function is:
s(t) = 0.5e^t - 0.5t - 0.5
3. Find the time at which the car has traveled 18m:
Set the position function equal to the distance traveled, s = 18, and solve for t:
0.5e^t - 0.5t - 0.5 = 18
0.5e^t - 0.5t = 18.5
We can solve this equation numerically, using methods such as Newton's method or a graphing calculator, to find the value of t.
Once we find the value of t, we can substitute it back into the velocity function to find the magnitude of the velocity, |v(t)|, and into the acceleration function to find the magnitude of the acceleration, |a(t)|.
Please note that the above process involves differential calculus and numerical methods, so be sure to review those concepts before attempting the calculations.
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