Video: Finding the Velocity of an Object given the Position as a Function of Time

The position of an object changes as a function of time according to 𝑥(𝑡) = −3𝑡² m. What is the object’s velocity when 𝑡 = 1 s? What is the object’s speed when 𝑡 = 1 s?

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Video Transcript

The position of an object changes as a function of time according to 𝑥 of 𝑡 equals negative three 𝑡 squared meters. What is the object’s velocity when 𝑡 equals one second? What is the object’s speed when 𝑡 equals one second?

Given a function describing an object’s position as a function of time, we want to know its velocity when 𝑡 is equal to one second and its speed at that same time. In other words, we want to solve for its instantaneous velocity and instantaneous speed. We can recall that an object’s instantaneous velocity is equal to the derivative of its position with respect to time. In our case, we’re given 𝑥 or position as a function of time and can plug in for that expression. When we take its derivative with respect to time, we find a result of negative six times 𝑡, now with units of meters per second. This is the instantaneous velocity of our object at a general time 𝑡. But we want to solve for that velocity when time is equal to one second. When 𝑡 is equal to one second, our instantaneous velocity is negative six times one meters per second, or negative six meters per second. That’s our instantaneous velocity when 𝑡 equals one second. Now what about its speed at that time?

We can recall that instantaneous speed is a scalar quantity. And it’s equal to the magnitude of instantaneous velocity. This means that the instantaneous speed of our object, when 𝑡 is equal to one second, is equal to the absolute value of negative six meters per second. This simplifies to six meters per second. That’s the object’s instantaneous speed when time equals one second.

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