Find an expression for the displacement?

Question

The velocity of a particle after #t# seconds is given by #v=(t-2)ms^-1#. After #6# seconds, the displacement is #11m# from #O#.

Find an expression for the displacement, #r#.

Find the displacement after #8# seconds.

Find the velocity when the particle has a displacement of #53m# from #O#.

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Answer 1 · 2018-03-10T01:20:56

Given expression for velocity of a particle by

#v=(t−2)\ ms^-1#

Comparing with kinematic expression

#v=u+at# ...............(1)

we get

#u=-2\ ms^-1and a=1\ ms^-2#

(1). Kinematic expression for displacement is

#s=s_0+ut+1/2at^2#

Inserting given conditions we get the expression for displacement #r# as

#r=r_0+(-2)t+1/2(1)t^2#
#r=r_0-2t+1/2t^2#

To find #r_0#, use the given condition at #t=6\ s#

#r(6)=11=r_0-2(6)+1/2(6)^2#
#=>11=r_0-12+18#
#=>r_0=5\ m#

#:.# Expression for #r# becomes

#r=5-2t+1/2t^2# ........(2)

(2). From (2)

#r(8)=5-2xx8+1/2(8)^2#
#=>r(8)=5-16+32#
#=>r(8)=21\ m#

(3). From (2)

#53=5-2t+1/2t^2#
#=>t^2-4t-96=0#

Solving the quadratic by split the middle term we get

#t=-8 and 12#,

Ignoring #-ve# root as time can not be negative and inserting in (1) we get

#v(12)=-2+1xx12=10\ ms^-1#