Find the time it takes the stone to catch up with the ball and find the velocities of the stone and the ball when they are at the same height.A...

The distance travelled by an an object thrown upwards is
given by the formula:

s = ut -
(gt^2)/2

Where:

s = Distance
travelled

u = Initial upward
speed

t = Time elapsed

g =
Acceleration due to gravity = 9.8 m/s^2

Using this formula
the distance travelled by the ball at time t is given
by:

s(ball) = 11t -
(gt^2)/2

As the stone is thrown one second after the ball,
the time applicable to the stone, as compared to that applicable to ball will be (t -
1).

Therefor the distance travelled by the stone at time t
is given by:

s(stone) = 25(t -1) - [g(t -
1)^2]/2

= 25t - 25 - g(t^2 - 2t +
1)/2

= 25t - 25 - (gt^2 - 2gt +
g)/2

= 25t - 25 - (gt^2)/2 + gt -
g/2

When the stone and ball are at the same
height:

s(ball) =
s(stone)

Substituting the above expressions of distance for
ball and stone in above equation:

11t - (gt^2)/2 = 25t - 25
- (gt^2)/2 + gt - g/2

==> 25t - 11t - 25 - (gt^2)/2
+ (gt^2)/2 + gt - g/2

==> 14t - 25 + gt - g/2 =
0

==> 14t + gt = 25 +
g/2

==> t(14 - g) = 25 -
g/2

==> t = (25 + g/2)/(14 +
g)

Substituting value of g in above
equation:

==> t = (25 + 9.8/2)/ (14 + 9.8) =
29.9/23.8 = 1.2563 s

Time taken by stone to catch up with
ball

= t - 1 = 1.2563 - 1 = 0.2563
s

Velocity of ball at 1.2563 seconds = 11 - 9.8*1.2563 = -
1.3118 m/s

Velovity of stone at 1.2563 seconds = 25 -
9.8*0.2563 = 22.4880
m/s

Answer:

Time taken by
stone to catch up with ball = 0.2563 s

Velocity of ball at
this time = - 1.3118 m/s

Velocity of stone at this time =
22.4880

Please not that negative velocity of the ball
indicates that when the stone reaches the same height as the ball, the is falling down
after reaching its highest point.