1. The position of a particle moving in a straight line is given by s = t 3 ? 5

Question

1. The position of a particle moving in a straight line is given by

s = t3 ? 5t2ft after t seconds. Find an expression for its acceleration a after a time t.

2. Suppose annual per capita sales of bottled water in a certain country in the period 2000-2010 could be approximated by

Q(t) = ?0.13t2 + 3t + 15 gallons    (0 ? t ? 10),

where t is time in years since 2000. According to the model, were annual per capita sales of bottled water accelerating or decelerating in 2003?

accelerating

How fast?
_______ gal/yr2

Explanation / Answer

1. Since s(t) is the position function, the acceleration a(t) will be the second derivative of s(t).

Differentiating twice, we get

a(t)= d2s/dt2 (t3 + 5t2)

= ds/dt (3t2 + 10t)

= 6t+10

Final answer:

a(t)= 6t+10 ft/s2

2. Again, the acceleration function is the 2nd derivative of Q(t). Therefore the acceleration is

d2Q/dt2 (0.13t2+3t+15)=

dQ/dt (0.26t+3)=

0.26

The acceleration is constant for all values of t.

Therefore the acceleration will be 0.26 gal/yr2

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