question_answer 76)

                  Figure shows \[({{\upsilon }_{x}},\,t),\] and \[({{\upsilon }_{y}},\,t)\] diagrams for a body of unit mass. Find the force a function of time.

Answer:

                  (1) Here \[{{\upsilon }_{x}}=2t\] between \[t=0\] and 1s                 \ \[{{a}_{x}}=2\]                 \[{{\upsilon }_{x}}=t\,\] between \[t=0\]and 1s.                 \[\therefore \]\[{{a}_{y}}=1\]                 \[\therefore \] \[\vec{a}\,={{a}_{x}}\,\hat{i}+{{a}_{y}}\hat{j}\,=2\hat{i}\,+\,\hat{j}\]                 \[\therefore \]\[\vec{F}=\,m\vec{a}=1\,(2\hat{i}+\hat{j})\,=2\hat{i}+j\,\]                 Between \[t=0\] and \[1s\]                 (2) \[{{\upsilon }_{x}}=2\,(2-t)\] between \[t=1\,s\] and \[t=2s\]                 \[\therefore \] \[{{a}_{x}}=-2\]                 \[{{\upsilon }_{y}}=\,1\,\] between \[t=1s\] and \[t=\,2s\]                 \[\therefore \] \[{{a}_{y}}=0\]                 \[\therefore \]\[\,\vec{a}\,={{a}_{x}}\hat{i}\,+{{a}_{y}}j=-2i\]                 \[\therefore \]\[\vec{F}=\,m\vec{a}\,=1\,(-2i)=-2\hat{i}\]                 For \[t>2s,\] both \[{{a}_{x}}\] and \[{{a}_{y}}\] are zero                              \[\therefore \]\[\vec{F}=0\]