To show that the two balls will meet each other between points L and M, we need to find the conditions under which their paths intersect. We can analyze the motion of the balls separately and set up equations for their positions.
Let's assume that the time taken for the balls to meet each other is "t" seconds after they are projected.
For the ball projected from point L:
Initial velocity (u) = 2u m/s
Acceleration due to gravity (g) = 9.8 m/s² (approximately)
Initial position (height above the ground) = 0 m
The equation for the vertical position of the ball projected from L at time "t" is given by:
hL(t) = 0 + (2u)t - (1/2)gt²
For the ball projected from point M:
Initial velocity (u) = u m/s
Acceleration due to gravity (g) = 9.8 m/s² (approximately)
Initial position (height above the ground) = d m
The equation for the vertical position of the ball projected from M at time "t" is given by:
hM(t) = d + ut - (1/2)gt²
Now, the balls will meet each other when their heights are the same (i.e., hL(t) = hM(t)):
0 + (2u)t - (1/2)gt² = d + ut - (1/2)gt²
Subtracting (1/2)gt² from both sides:
(2u)t = d + ut
Subtracting ut from both sides:
(2u - u)t = d
Simplifying:
ut = d
Now, we can determine the value of "t":
t = d/u
Now, let's find the conditions for which "t" is positive (i.e., the balls meet between L and M):
Since the ball from L is projected with a higher initial speed (2u), it will reach its maximum height first and then start descending. The ball from M is projected with an initial speed of u and will also reach its maximum height but at a slower rate. For the two balls to meet between L and M, "t" should be positive, which means the time it takes for them to meet should be greater than 0.
Therefore, we need to find the conditions for which d/u > 0.
Since distance (d) and initial speed (u) are both positive values, the condition for "t" to be positive is:
d/u > 0
This is always true because both d and u are positive, and any positive value divided by another positive value is still positive.
Next, let's find the conditions for which "t" is less than the time it takes for the ball from L to reach its maximum height.
The time taken by the ball from L to reach its maximum height (tmax) can be found using the equation:
hL(tmax) = 0
0 + (2u)tmax - (1/2)g(tmax)² = 0
Simplifying:
(2u)tmax = (1/2)g(tmax)²
(2u) = (1/2)gtmax
tmax = 2u/g
Now, for the balls to meet between L and M, "t" should be less than tmax:
d/u < tmax
Substituting the value of tmax:
d/u < 2u/g
Now, let's simplify this inequality:
dg < 2u²
Now, we want to find the conditions for which "u²" is less than or equal to (1/2)dg:
u² ≤ (1/2)dg
Finally, combining the conditions we found:
1/4gd < u² ≤ 1/2gd
So, the two balls will meet each other between points L and M if and only if 1/4gd < u² < 1/2gd.