Question 2
You MAY use a calculator to solve this problem.
A particle moves along the xaxis so that its acceleration at any time t > 0 is given by a(t) = 12t  18. At time t = 1, the velocity of the particle is v(1) = 0 and the position is x(1) = 9.
(a) Write an expression for the velocity of the particle v(t).
(b) At what values of t does the particle change direction?
(c) Write an expression for the position x(t) of the particle.
(d) Find the total distance traveled by the particle from t = 3/2 to t = 6.
Solution
(a) Answer v(t) = 6t^{2 } 18t + 6

We can find the equation of the v(t) by finding the antiderivative of a(t).
a(t) = 12t  18
v(t) = S (12t  18) dx
v(t) = 6t^{2}  18t + C_{o}

We can find C_{o} because we are given the information v(1) = 0 so that pins it down to one answer. Plug in the given values into the equation.
v(t) = 6t^{2}  18t +C_{o}
v(1) = 6(1)^{2 } 18(1) + C_{o }
0 = 6^{ } 18 + C_{o }
C_{o} = 0 + 18  6
C_{o }= 6

Therefore, the expression for v(t) is v(t) = 6t^{2 } 18t + 6
(b) Answer t = (3±√(5))
2

Since x'(t) = v(t), we can illustrate what's happening using a number line of v(t).
v(t) = 6t^{2 } 18t + 6
v(t) = 6 (t^{2 } 3t + 1),
+  +
3√(5) 3+√(5)
2 2
 And here we can see it graphically
(c) Answer x(t) = 2t^{3}  9t^{2} + 6t + 10

Like part (a) we can find the equation of the x(t) by finding the antiderivative of v(t), or differentiating a(t) twice.
x(t) = S (6t^{2}  18t +6)
= 2t^{3}  9t^{2} + 6t +C_{1}

We can find C_{1} given the information x(1) = 9.
x(t) = 2t^{3}  9t^{2} + 6t + C_{1}
x(1) = 2(1)^{3}  9(1)^{2} + 6(1) + C_{1}
9 = 2  9 + 6 + C_{1}
C_{1 }= 10
 Therefore, the expression for x(t) of the particle is x(t) = 2t^{3}  9t^{2} + 6t + 10
(d) Answer 153.8438 units.
 Integrating a positiontime graph within an interval will give you the total distance traveled for that specific time frame.
6
a = § x(t)
3/2
6
= § (2t^{3}  9t^{2} + 6t + 10) dx
3/2
6
= [(1/2)t^{4}  3t^{3} + 3t^{2} + 10t]
3/2
= [648  648 + 108 + 60]  [2.53125  10.125 + 6.75 + 15]
= 168  14.15625
= 153.8438 units
Comments (1)
Darren Kuropatwa said
at 11:16 pm on Jul 15, 2012
bedanidhi_kafley@yahoo.com writes: "I think there is a mistake in answer a) when finding Co ie v(t) = 6t2  18t +Co v(1) = 6(1)2  18(1) + Co 0 = 6  18 + Co Co = 0 + 18  6 Co = 6 there 186 is written 6 but i think it should be 12"
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