What is the average speed, on #t in [0,5]#, of an object that is moving at #12 m/s# at #t=0# and accelerates at a rate of #a(t) =5-3t# on #t in [0,4]#?
The acceleration of an object is simply the derivative of its velocity with respect to time, so the velocity of an object is the integral of the acceleration with respect to time, in this case
To get the average velocity, the simplest thing to do is to find the distance traveled in the specified time, which is the integral of the velocity between the limits:
The average velocity is then the total distance over the total time:
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To find the average speed, we need to calculate the total distance traveled and divide it by the total time taken. The distance traveled can be found by integrating the velocity function over the interval [0,5]. The velocity function is the integral of the acceleration function, so we integrate a(t) to get v(t), then integrate v(t) to get the distance function.
The velocity function v(t) = ∫(5 - 3t) dt = 5t - (3/2)t^2 + C
At t = 0, v(0) = 12, so C = 12.
Therefore, v(t) = 5t - (3/2)t^2 + 12
Now, integrate v(t) to get the distance function, s(t):
s(t) = ∫(5t - (3/2)t^2 + 12) dt = (5/2)t^2 - (1/2)t^3 + 12t + C
At t = 0, s(0) = 0, so C = 0.
Therefore, s(t) = (5/2)t^2 - (1/2)t^3 + 12t
Now, we find the total distance traveled by evaluating s(t) at t = 5:
s(5) = (5/2)(5)^2 - (1/2)(5)^3 + 12(5) = 125/2 - 125/2 + 60 = 60 meters
The average speed is then the total distance traveled divided by the total time taken, which is 5 seconds:
Average speed = Total distance / Total time = 60 meters / 5 seconds = 12 m/s
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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