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