AP calculus / Physics Question?

An object moves along the x-axis following the position function:

x(t) = - ( 3 / pi ) cos [ ( pi / 3 ) t ] + ( 3 / pi ) + 2

A) Are the statements both true, false or one true + one false? give your reasons

Statement 1: for 3 < t < 4.5, velocity is decreasing

Statement 2: for 3 < t < 4.5, the speed is increasing

I thought they were both true since velocity is becoming more negative:

t = 3, v = 0 then t = 3.5, v = -1m/s etc.

I think it is speeding up since the acceleration graph is all negative:

t = 3, a = -3m/s^2 then t = 4, a = -1 m/s^2

so am i right or am I wrong??

b) What is the total distance travelled in 4 s?

I thought you use the position function for intervals [0,3] since velocity is positive and [3,4] since velocity is negative...

so am i right???

please explain clearly

Update:

for part B), the object changes from positive velocity to negative velocity, so wouldn't that count as changing direction, so you use intervals 0,3 and 3,4???????

Update 2:

for part B), the object changes from positive velocity to negative velocity, so wouldn't that count as changing direction, so you use intervals 0,3 and 3,4???????

2 Answers

Relevance
  • Anonymous
    8 years ago
    Best Answer

    Differentiate the function to get...

    x'(t) = 3/π * π/3 * sin(πt/3)

    = sin(πt/3)

    Graph that function, and we have....

    →Decreasing at 3 < t < 4.5

    →Increasing at 3 < t < 4.5

    Good luck!

    Source(s): Σ
  • Dand
    Lv 5
    8 years ago

    x(t) = - ( 3 / pi ) cos [ ( pi / 3 ) t ] + ( 3 / pi ) + 2

    v = dx/dt = (-3/pi) (pi/3) [-sin(t pi/3) = sin (pi/3) t

    t = 3 ===> v = sin pi = 0

    t = 4.5===> v = sin 3pi/2 = -1 ===> decreasing = statement 1

    b)

    x(4) = - ( 3 / pi ) cos [ ( pi / 3 ) 4 ] + ( 3 / pi ) + 2 = 1.5/pi + 3/pi + 2 =

    = (4.5 + 2pi)/pi = 3.43

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