The problem states that we need to find the speed at which a ball must be projected vertically upward so that the distance traveled by it in the 5th second is equal to the distance traveled in the 6th second.


- The ball is projected vertically upward.
- We need to find the speed at which the ball is projected.
- We need to compare the distances traveled in the 5th and 6th seconds.
- The time intervals are consecutive seconds.


To solve this problem, we can use the equations of motion for vertically projected motion.

The equation to calculate the displacement (distance traveled) in a given time interval is:

s = ut + (1/2)at^2

Where:
- s is the displacement (distance traveled)
- u is the initial velocity (speed of projection)
- t is the time interval
- a is the acceleration (in this case, acceleration due to gravity)

We can calculate the displacements in the 5th and 6th seconds using this equation.

Displacement in the 5th second:
s5 = u * 5 - (1/2) * g * (5)^2

Displacement in the 6th second:
s6 = u * 6 - (1/2) * g * (6)^2

Given that the displacements in the 5th and 6th seconds are equal, we can set up the equation:

u * 5 - (1/2) * g * (5)^2 = u * 6 - (1/2) * g * (6)^2

Simplifying the equation:
5u - 25g = 6u - 36g

Rearranging the equation:
u = 11g

Substituting the value of acceleration due to gravity (g = 9.8 m/s^2), we get:
u = 11 * 9.8
u = 107.8 m/s

Therefore, the ball must be projected vertically upward with a speed of 107.8 m/s to ensure that the distance traveled in the 5th second is equal to the distance traveled in the 6th second.

However, the answer provided in the question is '49', which seems to be incorrect based on the given information and calculations.