Motion In A Straight Line

Given,

%5Cvec%7Ba%7D%3D%5Calpha%20%5Cvec%7Bv%7D

%5Cbegin%7Bgathered%7D%20a%3D%5Calpha%20v%20%5C%5C%20%5Cfrac%7Bd%20v%7D%7Bd%20t%7D%3D%5Calpha%20v%20%5Cend%7Bgathered%7D

On%20Integrating%20both%20sides%3A

%5Cbegin%7Baligned%7D%20%26%5Cbegin%7Bgathered%7D%20%5Cint%20%5Cfrac%7Bd%20v%7D%7Bv%7D%3D%5Cint%20%5Calpha%20d%20t%20%5C%5C%20%5Clog%20_e(v)%3D%5Calpha%20t%2Bc%20%5Cend%7Bgathered%7D%5C%5C%20%26%5Ctext%20%7B%20%7D%5C%5C%20%26%5Cbegin%7Bgathered%7D%20v%3De%5E%7B%5Calpha%20t%2Bc%7D%3De%5E%7B%5Calpha%20t%7D%20%5Ccdot%20e%5Ec%20%5C%5C%20%5Cfrac%7Bd%20S%7D%7Bd%20t%7D%3De%5E%7B%5Calpha%20t%7D%20%5Ccdot%20e%5Ec%20%5Cend%7Bgathered%7D%20%5Cend%7Baligned%7D

%5Ctext%20%7B%20Integrate%20to%20find%20displacement%3A%20%7D

%5Cbegin%7Baligned%7D%20%26%5Cint%20d%20S%3De%5Ec%20%5Cint%20e%5E%7B%5Calpha%20t%7D%20%5Ccdot%20d%20t%5C%5C%20%26%5BS%5D_x%5Ey%3D%5Cleft%5B%5Cfrac%7Be%5Ec%7D%7B%5Calpha%7D%20%5Ccdot%20e%5E%7B%5Calpha%20t%7D%5Cright%5D_x%5Ey%20%5Cend%7Baligned%7D

So%2C

%5Cbegin%7Bgathered%7D%20S_%7B12%7D%3D%5BS%5D_%7Bt_1%7D%5E%7Bt_2%7D%3D%5Cleft%5B%5Cfrac%7Be%5Ec%7D%7B%5Calpha%7D%20%5Ccdot%20e%5E%7B%5Calpha%20t%7D%5Cright%5D_%7Bt_1%7D%5E%7Bt_2%7D%20%5C%5C%20S_%7B12%7D%3D%5Cfrac%7Be%5Ec%7D%7B%5Calpha%7D%20%5Ccdot%5Cleft(e%5E%7B%5Calpha%20t_2%7D-e%5E%7B%5Calpha%20t_1%7D%5Cright)%20%5C%5C%20S_%7B13%7D%3D%5Cfrac%7Be%5Ec%7D%7B%5Calpha%7D%20%5Ccdot%5Cleft(e%5E%7B%5Calpha%20t_3%7D-e%5E%7B%5Calpha%20t_1%7D%5Cright)%20%5Cend%7Bgathered%7D

%5Cbegin%7Bgathered%7D%20%5Cfrac%7BS_%7B13%7D%7D%7BS_%7B12%7D%7D%3D%5Cfrac%7B%5Cfrac%7Be%5Ec%7D%7B%5Calpha%7D%20%5Ccdot%5Cleft(e%5E%7B%5Calpha%20t_3%7D-e%5E%7B%5Calpha%20t_1%7D%5Cright)%7D%7B%5Cfrac%7Be%5Ec%7D%7B%5Calpha%7D%20%5Ccdot%5Cleft(e%5E%7B%5Calpha%20t_2%7D-e%5E%7B%5Calpha%20t_1%7D%5Cright)%7D%20%5C%5C%20%5Cfrac%7BS_%7B13%7D%7D%7BS_%7B12%7D%7D%3D%5Cfrac%7B%5Cleft(e%5E%7B%5Calpha%20t_3%7D-e%5E%7B%5Calpha%20t_1%7D%5Cright)%7D%7B%5Cleft(e%5E%7B%5Calpha%20t_2%7D-e%5E%7B%5Calpha%20t_1%7D%5Cright)%7D%20%5Cend%7Bgathered%7D

Taking

e^{\alpha t_1}

common%2C

%5Cbegin%7Bgathered%7D%20%5Cfrac%7BS_%7B13%7D%7D%7BS_%7B12%7D%7D%3D%5Cfrac%7Be%5E%7Bt_1%7D%5Cleft(e%5E%7B%5Calpha%20t_3-t_1%7D-1%5Cright)%7D%7Be%5E%7Bt_1%7D%5Cleft(e%5E%7B%5Calpha%20t_2-t_1%7D-1%5Cright)%7D%20%5C%5C%20%5Cfrac%7BS_%7B13%7D%7D%7BS_%7B12%7D%7D%3D%5Cfrac%7B%5Cleft(e%5E%7B%5Calpha%20t_3-t_1%7D-1%5Cright)%7D%7B%5Cleft(e%5E%7B%5Calpha%20t_2-t_1%7D-1%5Cright)%7D%3D%5Cfrac%7B%5Cexp%20%5Cleft(%5Calpha%20t_3-t_1%5Cright)-1%7D%7B%5Cexp%20%5Cleft(%5Calpha%20t_3-t_1%5Cright)-1%7D%20%5Cend%7Bgathered%7D

Chapter Questions
3 MCQs