Position Functions and Instantaneous Velocity Video Lessons

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Problem: The position of a ball rolling in a straight line is given by x(t) = 2.1 − 3.7t + 3.3t2, where x is in meters and t in seconds. (a) Determine the position of the ball at t = 2.0 s. (b) What is the average velocity from t = 2.0 s to t = 5.0 s? (c) What is its instantaneous velocity at t = 5.0 s?

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FREE Expert Solution

We are asked to find the position, average velocity, and instantaneous velocity for a ball, given its position function.

Remember that to find an object's position at a certain time, we just put that value of t into the function and calculate the result (as long as everything is in the right units).

To calculate the average velocity, we use the formula

vavg=xt=x2-x1t2-t1

Anytime we're given a position, velocity, or acceleration function and asked to find one or more of the others, we know it's a motion problem with calculus. A PVA diagram like the one below can help remind you of the relationships between the three functions:

PVddtdtA

To get from position to velocity, we take the derivative of the position function.

v(t)=dx(t)dt

We'll also need to remember the power rule of derivation, which is:

ddt(tn)=ntn-1  (for example, ddtt3=3t2)

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Problem Details

The position of a ball rolling in a straight line is given by x(t) = 2.1 − 3.7t + 3.3t2, where x is in meters and t in seconds.
(a) Determine the position of the ball at t = 2.0 s.
(b) What is the average velocity from t = 2.0 s to t = 5.0 s?
(c) What is its instantaneous velocity at t = 5.0 s?

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