A ball is thrown vertically upwards such that the distance traveled by it in 5th and 6th sec is equal. Find the speed of the ball.
Answers
Answer:
A ball is thrown vertically up with certain initial velocity = u m/s (say). As it is travelling upwards it is acted upon by gravitational force acting downward, with an acceleration g m/s². It is given that it travels equal distances in its 5th second and 6th second. It is only possible for the body to travel equal distances in two consecutive seconds, when in the 5th second, it is travelling upwards, reaching the maximum height at the end of 5th second and then in the 6th second it starts to descend down.
So at end of t = 5 second, v = 0 m/s.
Using the equation v = u +a t, we have in this case a = -10 m/s², t = 5 s,
0 = u m/s - 10 m/s² × 5s, or u = 50 m/s.
The ball was projected upwards with a velocity u =50 m/s.
The highest point is reached after 5 s. Using the relation, s= u t + ½ at². In our case u= 50 m/s, a = - 10 m/s², t= 5 s,we get,
s = 50 m/s × 5 s - ½ × 10 m/s² × 5²= 250m - 125m= 125 m
We could have also obtained the same result using the equation: v² - u² = 2 a h, where v= 0 m/s at the highest point, u = 50 m/s ( upward direction taken positive), a= -10 m/s² ( acceleration due to gravity taken negative as it is directed downward), substituting various values we get,
0² - 50² = 2 (-10 ) h, or h= 125 m, same as obtained earlier, as it should.
Let us check if the body travels the same distance in 5th second when going up and in 6th second when coming down.
Distance travelled in nth second by a body travelling with initial velocity u, and acceleration a,
Xn = u × 1 s + ½ a ( 2 n- 1 )
DISTANCE TRAVELLED WHEN GOING UP:
In this case u = 50 m/s, a = - 10 m/s² n=5s
X5 = 50 +½ (-10)(9)= 50 -45 = 5 m
The ball covers 5 m in the upward direction in the 5th second.
DISTANCE TRAVELLED WHEN GOING DOWN:
After reaching the highest point the velocity of the ball is u = 0 m/s , a = 10 m/s² n=1 second ( it has just started its descent)
Xn = 0 + ½ ×10× (2×1–1)= 5×1 =5m.
So the highest point reached by the ball is 125 m from the point of its being thrown up.