Question Video: Determining the Acceleration given the Velocity Expression of a Particle Mathematics • Higher Education

A particle moves along the ๐‘ฅ-axis. When its displacement from the origin is ๐‘  m, its velocity is given by ๐‘ฃ = 4/(3 + ๐‘ ) m/s. Find the particleโ€™s acceleration when ๐‘  = 3 m.

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Video Transcript

A particle moves along the ๐‘ฅ-axis. When its displacement from the origin is ๐‘  metres, its velocity is given by ๐‘ฃ equals four over three plus ๐‘  metres per second. Find the particleโ€™s acceleration when ๐‘  is equal to three metres.

Here, we have a function for the velocity in terms of ๐‘ . Weโ€™re being asked to find the acceleration. Now, acceleration is defined as the change in velocity with respect to time or the derivative of ๐‘ฃ with respect to ๐‘ก. We canโ€™t easily differentiate ๐‘ฃ with respect to ๐‘ก though without performing an extra step. So weโ€™re going to use implicit differentiation. We differentiate both sides of our equation with respect to ๐‘ก. And on the left, we get d๐‘ฃ by d๐‘ก. On the right, we get d by d๐‘ก of four over three plus ๐‘ .

To make this easier, weโ€™re going to change this to four times three plus ๐‘  to the power of negative one. And then the derivative of this expression with respect to ๐‘ก is equal to the derivative of it with respect to ๐‘  times d๐‘  by d๐‘ก. We then use the general power rule on the extension of the chain rule. And we see that derivative of four times three plus ๐‘  to the power of negative one with respect to ๐‘  is negative four times three plus ๐‘  to the power of negative two. And so we see that d๐‘ฃ by d๐‘ก is equal to negative four times three plus ๐‘  to the power of negative two times d๐‘  by d๐‘ก. We can write three plus ๐‘  to the power of negative two as one over three plus ๐‘  squared. But we also know that d๐‘  by d๐‘ก is ๐‘ฃ. So we can see we have an expression for the acceleration in terms of ๐‘ฃ and ๐‘ .

We need to evaluate this when ๐‘  is equal to three. So letโ€™s begin by working out ๐‘ฃ when ๐‘  is equal to three. When ๐‘  is equal to three, ๐‘ฃ is equal to four over three plus three which simplifies to two-thirds. And when ๐‘  is equal to three, acceleration is therefore is negative four over three plus three all squared times two-thirds which is equal to negative two over 27 metres per square second.

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