The motion of a grizzly bear stalking its prey, walking left and right of a fixed point in feet per second, can be modeled easily by the motion of a particle moving left and right along the x-axis, according to the acceleration equation . (Assume that the origin corresponds to the fixed point, and that a positive value for position means that the bear is located to the right of the fixed point as we watch him from a safe location.)

If the bear's velocity is 1 ft/sec when t = 0,

(a) Find the velocity equation for the bear's motion.

(b) Determine how fast the bear was traveling when t = 7 seconds.

(c) In what direction (from our vantage point) is the bear traveling when t = 5 seconds?

(d) How far does the bear walk during the first 10 seconds?

You may use a graphing calculator for this activity · Difficulty:

 

Solution

(a) Since we're given the acceleration equation, and velocity is the antiderivative of acceleration, integrate a(t) and use the fact that to find the velocity equation:

(b) This question asks us to find v(7).

In other words, the bear was traveling -1.579 ft/sec to our left.

(c) Since v(5) is negative, the bear is traveling to the left.

(d) You must split the interval [0,10] into segments according to where v crosses the x-axis, because we must measure how far the bear traveled forward and backward separately. The velocity equation has only one root on [0,10], and it occurs at x = 1.17012095. To figure out the total distance traveled, we must calculate the area betweenv and the x-axis on [0,1.17012095] and [1.17012095,10]. However, to get the correct answer, both integrals must be positive, so I'll take the absolute value of the second integral, since that signed area falls below the x-axis and will automatically be negative: