Problem : Suppose a rock is thrown straight up from atop a 200 -meter-high cliff at an initial speed of 30 feet per second. The height, in meters, of the rock above the ground (until it lands) at time t is given by the function h(t) = - gt ^{2}/2 + 30t + 200 , where g 9.81 is a constant of gravitational acceleration. When does the rock reach its maximum height? What is this maximum height? How fast is the rock moving after 3 seconds?
When the rock reaches its maximum height, it is instantaneously stationary, with speed 0 . Solving
h'(t) = - gt + 30 = 0 |
h(30/g) = +30 +200 = +200 245.89 |
h'(3) = (- g)(3) + 30 0.58 |
Problem : The position of a box, in a certain coordinate system, attached to the end of a spring is given by p(t) = sin(2t) . What is the acceleration of the box at time t ? How does this relate to its position?
The velocity of the box is equal to
p'(t) = 2 cos(2t) |
p''(t) = - 4 sin(2t) = - 4p(t) |
Problem : Suppose the velocity of a sprinter (in meters per second) at time t seconds after the start of a 40 meter dash is given by
v(t) = 3 log(t + 1) |
v'(t) = |