1. The position function (in meters) of a particle moving along a
horizontal line in \(t\) seconds is given by \(s(t) = 2 t^3 - 3 t^2 - 12t +8\).
(a) Find the velocity and acceleration functions.
(b) At what time(s) is
the particle at rest?
(c) On what time interval(s) is the particle moving to
the right or left?
(d) Determine the time interval(s) when the particle is
speeding up or slowing down.
2. If a ball is thrown vertically upward with a velocity of 80 feet per second, its height after \( t \) seconds is \( s(t)=80t-16t^2 \). Answer the following questions.
(a) What is the velocity of the ball after 1 sec?
(b) What is the maximum height reached by the ball?
(c) What is the velocity of the ball when it is 96 feet above the ground on its way up?
(d) With what velocity does the ball hit the ground?
3. A particle moves according to a law of motion \( s(t)=t^3-12t^2+36t\), where \(t\) is measured in seconds and \(s\) in feet. Answer the following questions.
(a) What is the velocity of the particle after 3 sec?
(b) When is the particle at rest?
(c) When is the particle moving in the positive direction?
(d) Find the total distance traveled during the first 8 sec.
(e) What is the acceleration after 2 sec?
4. A slingshot launches a stone vertically from the top of a wall. Its
height (in feet) after $t$ seconds is given by \( s(t)= 100+112t-16t^2.\)
(a) Find the velocity of the stone after 2 seconds.
(b) Find the maximum height of the stone.
(c) Find the velocity of the stone after it has risen 260 feet.
5. Suppose the position of an object moving horizontally after \(t\)
seconds is given by the following function \(s=f(t)\), where \(s\) is measured
in feet, with \(s \ge 0\) corresponding to positions right of the origin.
\( f(t)= 24t-4t^2; 0 \le t \le 8 \)
(a) Find the velocity function. When is the object stationary, moving to
the right, and moving to the left?
(b) Determine the velocity and acceleration of the object at \(t=1\).
(c) Determine the acceleration of the object when its velocity is zero.
(d) On what intervals is the speed increasing?