BODY FALLING UNDER GRAVITY 191 



157. If the body is projected vertically upwards with velocity 

 u, we may measure the distance s vertically upwards, and the 

 acceleration in this direction will be g. Thus we have 



v = u gt, 

 2 gs u 2 v z , 



where s is the distance upwards described after time t, and v is 

 the upward velocity. From the first equation we see that s = 



not only when t = 0, but also when t = --- Thus the particle 



t/ 



returns to its original position after time 2 u/g. When s = the 

 third equation shows that u 2 = v*. Thus when the particle returns, 

 its velocity is the same as when it started. Clearly this must be 

 S0j for the potential energy is the same, and therefore the kinetic 

 energy also is the same. 



EXAMPLES 



1. If an express train is doubled, the first half being given 5 minutes' start, 

 and attaining its maximum booked speed of 48 miles an hour after moving with 

 constant acceleration for a mile, prove that the two halves will run about 4 

 miles apart, but the first half will have gone 3 miles before the start of the 

 second half. 



\>/2. A train passes another on a parallel track, the former having a velocity 

 of 45 miles an hour and an acceleration of 1 foot per second per second, the 

 latter a velocity of 30 miles an hour and an acceleration of 2 feet per second 

 per second. How soon will the second be abreast of the first again, and how 

 far will the trains have moved in the meantime ? 



3. A body is dropped from a balloon at a height of 70 feet from the ground. 

 Find its velocity on reaching the ground, if the balloon is (a) rising, (&) falling, 

 with a velocity of 30 feet a second. 



(y/4. A stone is dropped into a well, and the sound of the splash reaches the 

 top after 9 seconds. Find the depth of the well, the velocity of sound being 

 1100 feet per second. 



1 5. An elevator descends with an acceleration of 5 feet per second per second 

 until its velocity is 20 feet per second, after which its velocity remains uniform. 

 After it has been in motion for 6 seconds, a stone is dropped on to it from the 

 point at which the elevator started. How soon will it strike the elevator ? 



6. A juggler keeps three balls going with one hand, so that at any instant 

 two are in the air and one in his hand. If each ball rises to a height of 4 feet, 

 show that the time during which a ball stays in his hand is second. 



