A ball, dropped vertically, falls d metres in t seconds. d is directly proportional to the square of t. The ball drops 45 metres in the first 3 seconds. How far does the ball drop in the next 7 seconds?

If d is directly proportional to the square of t, we write this as d= kt² , where k = the proportionality constant which we must find. Substitute in the values given in the question into our equation to get 45= k (3²) and hence 45= 9k , so k=5. Now we know the value of k, we can find out how far the ball travels in a further seven seconds. We cant isolate these seven seconds using our equation, so instead use the value t=10 (3+7 seconds), to find d after 10 seconds of falling. d= 5(10²) = 500. If the ball has gone 500m after 10 seconds, and we know it had travelled 45 of those metres in the first 3 seconds, we find how far it went in the 7 seconds after by doing 500 - 45 = 455 m.