The position of an object moving along a line is given by #p(t) = t - tsin(( pi )/3t) #. What is the speed of the object at #t = 3 #?

1 Answer
Dec 7, 2016

Answer:

#1 + pi#

Explanation:

Velocity is defined as
#v(t) -= (dp(t))/dt#

Therefore, in order to find speed we need to differentiate function #p(t)# with respect to time. Please remember that #v and p# are vector quantities and speed is a scalar.

#(dp(t))/dt = d/dt(t - t sin(pi/3 t))#
#=>(dp(t))/dt = d/dtt - d/dt(t sin(pi/3 t))#

For the second term will need to use the product rule and chain rule as well. We get

#v(t) = 1 - [t xxd/dtsin(pi/3 t)+sin(pi/3 t) xxd/dt t]#
#=>v(t) = 1 - [t xxcos(pi/3 t)xxpi/3+sin(pi/3 t)]#
#=>v(t) = 1 - [pi/3t cos(pi/3 t)+sin(pi/3 t)]#

Now speed at #t=3# is #v(3)#, therefore we have

#v(3) = 1 - [pi/3xx3 cos(pi/3 xx3)+sin(pi/3 xx3)]#
#=>v(3) = 1 - [pi cos(pi)+sin(pi)]#

Inserting values of #sin and cos# functions
#v(3) = 1 - [pixx(-1) +0]#
#v(3) = 1 + pi#