Video Transcript
Complete the following. If the work done by a force at time
𝑡 seconds is given by 𝑤 of 𝑡 is equal to two 𝑡 squared plus three 𝑡 plus nine
joules, the power of this force at 𝑡 equals five seconds equals what.
In this question, we’re given an
expression for the work done by a force in terms of time 𝑡. This is equal to two 𝑡 squared
plus three 𝑡 plus nine and is given in the standard units of joules. We know that the power supplied by
a force is the time derivative of the work done. This means that we can find an
expression for the power by differentiating the expression for work done.
The power rule of differentiation
tells us that if 𝑦 is equal to 𝑎 multiplied by 𝑥 to the power of 𝑛, then d𝑦 by
d𝑥 is equal to 𝑛 multiplied by 𝑎 multiplied by 𝑥 to the power of 𝑛 minus
one. We can use this to differentiate
our expression term by term.
Differentiating two 𝑡 squared
gives us two multiplied by two multiplied by 𝑡 to the power of two minus one. This is equal to four 𝑡. As three 𝑡 can be written as three
𝑡 to the power of one, differentiating this gives us one multiplied by three
multiplied by 𝑡 to the power of one minus one. This is equal to three multiplied
by 𝑡 to the power of zero. And since anything to the power of
zero is one, this leaves us with the constant three. Finally, the derivative of any
constant is equal to zero.
𝑝 of 𝑡, the power of the force,
is therefore equal to four 𝑡 plus three. We want to calculate this when 𝑡
is equal to five seconds. Substituting 𝑡 equals five into
our expression gives us 𝑝 of five is equal to four multiplied by five plus
three. This is equal to 23. And we can therefore conclude that
the power of the force at 𝑡 equals five seconds is equal to 23 watts.
