Video: Finding the Velocity of a Particle from the Expression of Position with Time

A particle started moving in a straight line. After 𝑡 seconds, its position relative to a fixed point is given by 𝑟 = (𝑡² + 4𝑡 − 1) m, 𝑡 ≥ 0. Find the velocity of the particle when 𝑡 = 5 s.

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Video Transcript

A particle started moving in a straight line. After 𝑡 seconds, its position relative to a fixed point is given by 𝑟 is equal to 𝑡 squared plus four 𝑡 minus one metres, where 𝑡 is greater than or equal to zero. Find the velocity of the particle when 𝑡 is equal to five seconds.

Here, we’ve been given the position of a particle relative to a fixed point at a time 𝑡 seconds. This is a function of time. And this means the particle won’t necessarily travel at a constant velocity. We therefore need to find an expression for the velocity of the particle at a given time 𝑡 seconds.

So let’s recall what we actually mean by the word velocity. It’s the change in the particle’s displacement over time. This means we can find a function for the velocity by differentiating the function for the displacement or the position of the particle relative to the fixed point with respect to time.

So how do we differentiate the expression 𝑡 squared plus four 𝑡 minus one? We multiply each term by its power. And then we reduce that said power or exponent by one. So 𝑡 squared differentiates to two multiplied by 𝑡 to the power of one. That’s just two 𝑡. And four 𝑡 differentiates to one multiplied by four 𝑡 to the power of zero. And 𝑡 to the power of zero is one. So this differentiates to four. And the constant differentiates to zero. And this is because it’s currently negative one multiplied by 𝑡 to the power of zero. When we multiplied by that power of zero, we just get zero.

So we can say that the velocity at 𝑡 seconds is given by the expression two 𝑡 plus four. And since the displacement was in metres and the time is in seconds, we say that the velocity is two 𝑡 plus four metres per second. To find the velocity when 𝑡 is equal to five seconds, we’ll substitute five into this expression. When 𝑡 is equal to five, 𝑣 is equal to two multiplied by five plus four. Two multiplied by five is 10.

So the velocity is given by 14 metres per second.

And at this stage, it’s useful to remind ourselves of a graphic that can help us remember how to relate displacement, velocity, and acceleration. We’ve already seen that we can differentiate an expression for displacement with respect to time to find an expression for the velocity.

Similarly, we can differentiate an expression for the velocity with respect to time to form an expression for the acceleration. And since integration is the opposite of differentiation, we can find an expression for the velocity by integrating the expression for the acceleration with respect to time. And finally, we can integrate the expression for velocity with respect to time to find an expression for the displacement 𝑟.

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