Two grenades, A and B, are thrown horizontally with different speeds from the top of a cliff 70 m high. The speed of A is 2.50 m/s and the speed of B is 3.40 m/s. Both grenades remain in air for 3.77 s. Assume that the acceleration due to gravity is 9.86 m/s^2. What is the distance between A and B if they are thrown along the same straight line?

To find the horizontal distance between grenades A and B, we need to calculate the horizontal displacement of each grenade.

First, let's calculate the horizontal displacement of grenade A:
Since grenade A is thrown horizontally, there is no initial vertical velocity.
The vertical displacement is given by the formula: (vertical displacement) = (initial vertical velocity) * (time) + (1/2) * (acceleration due to gravity) * (time^2)
Here, the vertical displacement is -70 m (negative because the motion is downward), the initial vertical velocity is 0 m/s, the time is 3.77 s, and the acceleration due to gravity is -9.86 m/s^2 (negative because it acts downward).
So, -70 = 0 * 3.77 + (1/2) * (-9.86) * (3.77)^2
Simplifying, we get:
-70 = 0 - 17.5 * 14.1929
-70 = -249.57

Next, let's calculate the horizontal displacement of grenade B:
Grenade B is also thrown horizontally, so the initial vertical velocity is 0 m/s.
Using the same formula as above:
-70 = 0 * 3.77 + (1/2) * (-9.86) * (3.77)^2
Simplifying, we get:
-70 = 0 - 17.5 * 14.1929
-70 = -315.56

The horizontal displacements of both grenades are equal, but opposite in sign since one moves to the left and the other moves to the right. Therefore, the distance between A and B is the sum of their horizontal displacements:
Distance between A and B = |-249.57| + |-315.56|
Distance between A and B = 249.57 + 315.56
Distance between A and B = 565.13 m

So, the distance between grenades A and B is 565.13 m.

To find the distance between grenades A and B, we need to analyze their horizontal motion. Since both grenades are thrown horizontally, the vertical component of their motion is not affected by their initial speeds. Therefore, we don't need to consider the vertical motion in this problem.

Let's start by calculating the horizontal distance traveled by each grenade using their respective speeds and the time of flight.

For grenade A:
Horizontal distance (A) = Speed (A) * Time (A)
Horizontal distance (A) = 2.50 m/s * 3.77 s

For grenade B:
Horizontal distance (B) = Speed (B) * Time (B)
Horizontal distance (B) = 3.40 m/s * 3.77 s

Now, to find the distance between grenades A and B, we can subtract the horizontal distance traveled by A from the horizontal distance traveled by B.

Distance between A and B = Horizontal distance (B) - Horizontal distance (A)
Distance between A and B = (3.40 m/s * 3.77 s) - (2.50 m/s * 3.77 s)

Calculating this expression, we get:

Distance between A and B = 12.81 m - 9.43 m
Distance between A and B = 3.38 m

Therefore, the distance between grenades A and B, when thrown along the same straight line, is approximately 3.38 meters.

The horizontal distances are given by

Da(t) = 2.5t m
Db(t) = 3.4t m
Distance between A and B after 3.77 seconds is therefore
Distance = Db(3.77)-Da(3.77)
assuming the grenades are thrown in the same straight line in the same direction.
The acceleration due to gravity has no bearing on the distance between A and B, if air resistance can be ignored.