An object's velocity is given by #v(t) = (2t^2 +t +1 , sin2t )#. What is the object's rate and direction of acceleration at #t=6 #?
We're asked to find the the magnitude (rate) and direction of an object's acceleration at a given time, given its velocity equation.
To do this, we need too differentiate the velocity component equations, which are given as
Finding the derivatives:
The magnitude of the acceleration is thus
And the direction is
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The object's acceleration is (4, 4). Its rate of acceleration is 4 units per second squared, and its direction is opposite to the direction of the velocity vector.
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To find the object's rate and direction of acceleration at ( t = 6 ), we need to calculate the second derivative of the position function ( v(t) ) with respect to time. The second derivative gives us the acceleration vector.
 First, find the first derivative of ( v(t) ) with respect to time ( t ) for both components:
[ v(t) = (2t^2 + t + 1, \sin(2t)) ]
[ v'(t) = (4t + 1, 2\cos(2t)) ]
 Then, find the second derivative of ( v(t) ) with respect to time ( t ) for both components:
[ v''(t) = (4, 4\sin(2t)) ]
 Evaluate the acceleration vector ( v''(6) ) at ( t = 6 ):
[ v''(6) = (4, 4\sin(12)) ]

Calculate the magnitude of acceleration: [ v''(6) = \sqrt{4^2 + (4\sin(12))^2} ]

Determine the direction of acceleration: [ \text{Direction of acceleration} = \tan^{1}\left(\frac{4\sin(12)}{4}\right) ]

Calculate the numerical values: [ v''(6) \approx \sqrt{16 + 16\sin^2(12)} ] [ v''(6) \approx \sqrt{16 + 16(0.2108)} ] [ v''(6) \approx \sqrt{16 + 3.369} ] [ v''(6) \approx \sqrt{19.369} ] [ v''(6) \approx 4.4 , \text{(approximately)} ]
[ \text{Direction of acceleration} \approx \tan^{1}\left(\frac{4\sin(12)}{4}\right) ] [ \text{Direction of acceleration} \approx \tan^{1}(0.688) ] [ \text{Direction of acceleration} \approx 35.05^\circ ]
Therefore, at ( t = 6 ), the object's rate of acceleration is approximately ( 4.4 ) and its direction of acceleration is approximately ( 35.05^\circ ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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