A particle moves along the 𝑥-axis
such that at time 𝑡 seconds, its displacement from the origin is given by 𝑠 is
equal to 𝑡 cubed minus 4𝑡 squared metres, where 𝑡 is greater than or equal to
zero. What is the particle’s average
velocity in the first 10 seconds?
Our first thought in a question
like this might be to differentiate the expression for displacement to get an
expression for velocity. However, in this question, we need
to calculate the average velocity. This can be calculated by dividing
the total displacement by the total time. We need to calculate the average
velocity in the first 10 seconds. This means that the total time is
Initially, when 𝑡 equals zero, the
displacement is equal to zero cubed minus four multiplied by zero squared. This is equal to zero metres as the
particle starts at the origin. At 𝑡 equals 10, then the
displacement is equal to 10 cubed minus four multiplied by 10 squared. 10 cubed is equal to 1000. 10 squared is equal to 100. And multiplying this by four gives
us 400. 1000 minus 400 is equal to 600. Therefore, the displacement at time
10 seconds is 600 metres.
We can calculate the total
displacement by subtracting these two numbers: 600 metres and zero metres. This gives us 600 metres. The average velocity can,
therefore, be calculated by dividing 600 by 10. This is equal to 60. The displacement is measured in
metres and the time in seconds. Therefore, the velocity will be
measured in metres per second.
We can, therefore, conclude that
the average velocity of the particle in the first 10 seconds is 60 metres per