48.] RECTILINEAR MOTION. 2 



(n) A train of 120 tons is running 25 miles an hour. Find what 

 constant force is required to bring it to rest : (a) in 3 min. ; (b) in 

 half a mile. 



(12) If it takes i min. to coast down a hill on a uniformly sloping 

 road of i mile length, and the coefficient of friction be 0.02, what is the 

 height of the hill ? 



47. Kinetic Energy and Work. The dynamical equation of 

 motion, (i), Art. 38, can often be integrated after multiplying 

 both members by vdt=ds\ this makes the left-hand member an 

 exact differential, viz. the differential of the kinetic energy \ mv* 1 , 

 while the right-hand member, Fds t represents the elementary 

 work done by the force F: 



If F be given as a function of s alone, this equation can be inte- 

 grated, say from the time t Q to the time /. Denoting by s^ 

 and VQ the values of s and v at the time / (Fig. 7), we find 



I 



/* 



Fds. (12) 



This equation gives the velocity v as a function of the distance 

 s, counted from the arbitrary origin O. As v = ds/dt, a second 



t , V t, V 



fc 



U 



-s=s- 

 Fig. 7. 



integration will give s as a function of t. Examples of this 

 method have been given in Part I., Arts. 109, 117, 119, 121, 

 122 ; it will here only be necessary to call attention to the 

 dynamical meaning of the quantities involved. 



48. The left-hand member of equation (12) represents evi- 

 dently the increase in the kinetic energy of the moving particle, 

 while the right-hand member expresses the work done by the 

 force .F. during the passage of the particle from the point P Q 

 to the point P (see Part II., Arts. 71, 72). Hence, the meaning 



