12 KEST AND MOTION 



8. A ship whose head is pointing due south is steaming across a current run- 

 ning due west ; at the end of two hours it is found that the ship has gone 36 miles 

 in the direction 15 west of south. Find the velocities of the ship and current. 



9. A person traveling eastward at the rate of 3 miles an hour finds that the 

 wind seems to blow directly from the north ; on doubling his speed it appears 

 to come from the northeast. Find the direction of the wind and its velocity. 



ACCELERATION 



11. Acceleration is rate of increase of velocity. If we find that 

 the velocity of a moving point increases by an amount / in a sec- 

 ond, no matter which second is selected, we say that the motion 

 of the point has a uniform acceleration / per second. For instance, 

 a stone or other body falling under gravity is found to increase 

 its velocity by a certain constant velocity f per second, where f 

 denotes a velocity of about 32 feet per second. Thus we say that 

 a falling stone has a uniform acceleration of / per second, or of 

 about 32 feet per second per second. 



Generally, however, an acceleration will not be uniform ; the 

 rate of increase of velocity will be different at different stages of 

 the journey. To find the acceleration at any instant, we observe 

 the change in velocity during an infinitesimal interval dt of time. 



If dv is the increase of velocity, we say that is the acceleration 



Cut 



at the instant at which dt is taken. An acceleration will of course 

 have sign as well as magnitude, for the velocity may be either 

 increasing or decreasing. When the velocity is decreasing, the 

 acceleration is reckoned with a negative sign. A negative accelera- 

 tion is spoken of as a retardation. Thus a retardation / means 

 that the velocity is diminished by an amount / per unit of time. 



EXAMPLES 



1. A workman fell from the top of a building and struck the ground in 

 4 seconds. With what velocity did he strike the ground, the acceleration due to 

 gravity being 32 feet per second per second ? 



2. A train has at a given instant a velocity of 30 miles an hour, and moves 

 with an acceleration of 1 foot per second per second. Find its velocity after 

 20 seconds. 



