# Question Video: Finding the Times when the Velocity of a Particle Equals Zero Mathematics • Higher Education

A particle moves along the π₯-axis. Its position from the origin is π  meters at a time π‘ seconds. The position is given by the equation π  = 2 sin ((5π/7)π‘) + 7. Find the times π‘ at which the particleβs velocity is equal to 0.

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### Video Transcript

A particle moves along the π₯-axis. Its position from the origin is π  meters at a time π‘ seconds. The position is given by the equation π  is equal to two sin of five π over seven π‘ plus seven. Find the times π‘ at which the particleβs velocity is equal to zero.

In this question, we will begin by finding an expression for the particleβs velocity at time π‘. We are told that the displacement from the origin at time π‘ is equal to two sin of five π over seven π‘ plus seven. We recall that π£ is equal to dπ  by dπ‘. We can differentiate our expression for the displacement π  with respect to π‘ to find an expression for the velocity π£. Our expression for π  contains the sine function. And we know that if π¦ is equal to sin of ππ₯, then dπ¦ by dπ₯ is equal to π multiplied by cos of ππ₯.

This means that in order to differentiate the first term, we begin by multiplying five π over seven by two. Differentiating two sin of five π over seven π‘ with respect to π‘ gives us 10π over seven cos of five π over seven π‘. Differentiating a constant gives us zero. So this is our expression for dπ  by dπ‘ or the velocity π£.

We are asked to find the times π‘ at which the velocity is equal to zero. In order to do this, we set the expression 10π over seven multiplied by cos of five π over seven π‘ equal to zero. We can divide both sides by 10π over seven such that cos of five π over seven π‘ equals zero. Next, we recall that if cos π equals zero, π is equal to π over two plus ππ where π is an integer. In our question, cos of five π over seven π‘ equals zero. And this means that five π over seven π‘ must be equal to π over two plus ππ.

We can simplify this equation by dividing through by π. This means that five-sevenths π‘ is equal to one-half plus π. We can then divide through by five-sevenths, which is the same as multiplying by seven-fifths giving us π‘ is equal to seven-tenths plus seven-fifths π. The times π‘ at which the particleβs velocity is equal to zero are π‘ is equal to seven-tenths plus seven-fifths π, where π is an integer.