Partial horizontal motion

Question

given x(t) = (3t)/pi + cos(t). find the particle's horizontal position when its acceleration is -1?

Answer 1

Acceleration is the second derivative of position.

Once we calculate the acceleration x"(t), we can use it to see for what values of t the acceleration is -1.

Those values of t then give us the sought positions x(t).

x(t) = 3t/π + cos(t)

x'(t) = 3/π - sin(t)

x"(t) = -cos(t)

Now to get the values of t where acceleration is -1:

-cos(t) = -1

t = 2kπ for any integer k

In other words, acceleration is -1 for t = 0, 2π , 4π , ...

At those times the horizontal position:

x(2kπ) = 2(2kπ)/π + cos(π) = 4k + 1

In other words, the particle will have acceleration -1 when it is in horizontal positions

1, 5, 9, 13, ...