EXAMPLES XXIV 445 



(10) Find A + B + C. 



(11) Find B + C - A. 



(12) Find A + B - C. 



(13) Working with the values given in Example 1 of paragraph 

 213, find the acceleration when / = 10-03. 



(14) Working with the values given in Example 2 of paragraph 

 213, find the acceleration when / = 5*4. 



(15) A body is moving in such a way that the velocity at any 

 instant is given by v = a -f bt, and the direction is given by 

 (radians) = a -f $t, where t is time in seconds and a, b, a and (J 

 are constants. Find the magnitude and direction of the accelera- 

 tion at any instant. 



(16) A vector a is changing in direction and magnitude ; what 



is -T- if t is time ? Illustrate this by one example, say, by centri- 



HI 



petal acceleration of a point moving with constant speed in a 

 circular path. (B. of E., 1909.) 



(17) A body is moving in such a way that the velocity at any 

 instant is given by v = a 4- bt, and the direction is given by 

 (radians) = (3/ 2 , where t is time in seconds and a, b, and (3 are con- 

 stants. Find the magnitude and direction of the acceleration at 

 any instant. 



(18) The scalar product of two vectors is 12-74 and the vector 

 product is 15-76. Find the angle between the vectors. If the 

 magnitude of one of the vectors is 5-6, what is the magnitude of 

 the other ? 



(19) Fifty pounds of shot per second moving horizontally with 

 a velocity of 2,500 feet per second due north strike an armour- 

 plate and leave the plate horizontally with a velocity of 800 feet 

 per second due east. What force is exerted on the plate ? Note 

 that momentum and force are vectors. 



Force is rate of change of momentum per second. 



Momentum is mass multiplied by velocity. 



The mass of 50 Ib. of shot is 50 -f- 32-2. (B. of E., 1906.) 



(20) An aeroplane which is propelled at a speed of 50 miles per 

 hour relatively to the air, is steered in a circular course during a 

 steady wind of 15 miles per hour from the south. What are the 

 actual speeds of the aeroplane when going north, south, east, and 

 west ? 



(21) Rework Question 20. For a steady wind of 20 miles per 

 hour from the north-east. 



