A particle moves along a straight line. For 0 ≤ t ≤ 5, the position of the particle is given by – . s(t)=3t+4t^2-t^3.

Question

A particle moves along a straight line. For 0 ≤ t ≤ 5, the position of the particle is given by – .

s(t)=3t+4t^2-t^3.

a) What is the average velocity of the particle over the interval 0 ≤ t ≤ 4

b) Find a general expression for the velocity of the particle as a function of time, v(t), and determine

the value of v at t = 2. Is the particle traveling to the left or the right at this point? Justify your answer.

c) Find all times t during the interval 0 ≤ t ≤ 5 that the particle changes direction. Justify your answer.

d) At t = 4, is the speed of the particle increasing or decreasing? Justify your answer.

Answer 1

a) Find the slope of the secant line of s(t) that is form from t=0 and t=5. that slope is the average velocity.

b) take the derivative of s(t) to get v(t). pfind v(2), if your answer is positive its moving to right if v(2) is negative then its moving to the left.

c) If the particle changes direction, then at some point the derivative, v(t), had to be equal to zero. therefore, find when v(t) =0 and then use a sign chart to test points to the left and to the right of the critical points you just found. if you get different signs to left and right of critical point then it means the particle did change direction.

d) is the speed increasing or decreasing at t=4? find the derivative of v(t), a(t). the acceleration function will tell you whether or not the particle's speed is increasing or decreasing. if a(4) is positive then the speed is increasing, if a(4) is negative then the speed is decreasing.

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