An object at rest begins moving

Question

An object at rest begins moving

horizontally on the x-axis, starting at position x(t) = 0 feet at time t = 0 seconds.Suppose, after exactly 2 seconds we measure the position of the
object to be 3 feet to the right of the origin AND measure the object to be moving to the leftward at a speed of 8 feet per second.

Caution: We do NOT know anything about the position or motion of the object, other than what we know at times t = 0
seconds and t = 2 seconds. So do NOT simply assume the object is moving at constant speed the whole time.

Goal of the Problem: Estimate the position of the object after 2.5 seconds:

(1) by writing a solution as if you were carefully and clearly
explaining it (without calculus) to someone who has NOT had
calculus AND

(2) by writing a solution (with calculus) using the ideas of
linearization.

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1 answer
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Answer 1

Explain Bot answered
1 year ago

(1) To estimate the position of the object after 2.5 seconds without using calculus, we can make use of the information we have at times t = 0 seconds and t = 2 seconds.

From the given information, we know that at t = 0 seconds, the object is at position x(t) = 0 feet. At t = 2 seconds, the object is 3 feet to the right of the origin and moving to the left at a speed of 8 feet per second.

Since we want to estimate the position at t = 2.5 seconds, we can assume that the object continues moving with the same speed and direction. Therefore, we can estimate that at t = 2.5 seconds, the object will have moved half of the distance it traveled between t = 0 seconds and t = 2 seconds.

The distance between t = 0 seconds and t = 2 seconds is 3 feet (since the object is 3 feet to the right of the origin at t = 2 seconds). Therefore, the object will have moved 1.5 feet at t = 2.5 seconds.

Since at t = 2 seconds, the object is moving to the left at a speed of 8 feet per second, we can estimate that at t = 2.5 seconds, the object will have moved another 0.5 seconds in the leftward direction. This means the object will have moved 0.5 * (-8) = -4 feet at t = 2.5 seconds.

Therefore, our estimate for the position of the object at t = 2.5 seconds is 1.5 feet to the left of the origin, which is at position x(t) = -1.5 feet.

(2) To solve the problem using calculus and the idea of linearization, we can start by finding the average rate of change of position between t = 0 seconds and t = 2 seconds.

The average rate of change of position is given by the formula:
average rate of change = (change in position) / (change in time)

Between t = 0 seconds and t = 2 seconds, the change in position is 3 feet (since the object is 3 feet to the right of the origin at t = 2 seconds). The change in time is 2 seconds.

Therefore, the average rate of change of position is 3 feet / 2 seconds = 1.5 feet per second.

Next, we can use this average rate of change to estimate the instantaneous rate of change of position at t = 2 seconds.

Since the object is moving to the left at a speed of 8 feet per second at t = 2 seconds, we can estimate the instantaneous rate of change of position as -8 feet per second.

Using the idea of linearization, we can approximate the change in position between t = 2 seconds and t = 2.5 seconds by multiplying the instantaneous rate of change of position (-8 feet per second) by the change in time (0.5 seconds):

change in position = (-8 feet per second) * (0.5 seconds) = -4 feet

Therefore, our estimate for the position of the object at t = 2.5 seconds is obtained by adding this change in position to the position at t = 2 seconds:

position at t = 2 seconds + change in position = 3 feet - 4 feet = -1 feet.

So, our estimate for the position of the object at t = 2.5 seconds is x(t) = -1 feet.