Explanation:

Let v be the initial velocity with which the ball is projected vertically upward.

The velocity of the ball at any time t after projection is given by the formula:

v = u + gt

where u is the initial velocity, g is the acceleration due to gravity, and t is the time elapsed since the ball was projected.

Distance travelled in nth second:

The distance travelled by the ball in the nth second is given by the formula:

Sn = u + (n-1/2)g

where n is the number of seconds elapsed since the ball was projected.

Therefore, the distance travelled by the ball in the 5th second is:

S5 = u + (5-1/2)g
= u + 4.5g

and the distance travelled by the ball in the 6th second is:

S6 = u + (6-1/2)g
= u + 5.5g

Equating the distances:

Since the distance travelled in the 5th second is equal to the distance travelled in the 6th second, we have:

S5 = S6

u + 4.5g = u + 5.5g

g = u

Substituting the value of g:

Substituting the value of g = u in the formula for S5, we get:

S5 = u + 4.5u
= 5.5u

Using the formula:

We know that the distance travelled by the ball in the 5th second is 5.5 times its initial velocity. Therefore, we have:

5.5u = v(5) - (1/2)g(5)^2

where v(5) is the velocity of the ball at the end of the 5th second.

Substituting the value of g = u, we get:

5.5u = v(5) - (1/2)u(5)^2
= v(5) - 12.5u

Simplifying, we get:

v(5) = 18u

Similarly, we can find the velocity of the ball at the end of the 6th second as:

v(6) = 22u

Equating the distances again:

Since the distance travelled in the 5th second is equal to the distance travelled in the 6th second, we have:

S5 = S6

5.5u = v(6) - (1/2)g(6)^2
= 22u - 18u
= 4u

Simplifying, we get:

u = 49 m/s

Therefore, the ball must be projected vertically upward with an initial velocity of 49 m/s.