a stone is from the top of a tower 300m high and at the same time another stone is projected vertically upward with initial velocity 75m/s. calculate when and where the two stone meet
Solve for the t when
first stone height = second stone height
300 - 4.9 t^2 = 75 t- 4.9 t^2
The "4.9" is g/2, in m/s^2
75 t = 300
t = 4 seconds
height = 300 - 4.9*16 = 226 m
To find out when and where the two stones meet, we can start by analyzing the motion of each stone separately.
Let's consider the stones as Stone A (dropped from the tower) and Stone B (projected upward).
For Stone A:
- Initial velocity (u) = 0 m/s (as it is dropped)
- Acceleration (a) = 9.8 m/s² (acceleration due to gravity, acting downward)
For Stone B:
- Initial velocity (u) = 75 m/s (projected upward)
- Final velocity (v) = 0 m/s (at the highest point, where it changes direction)
- Acceleration (a) = -9.8 m/s² (acceleration due to gravity, acting downward)
We will use the kinematic equations to solve for the time and height at which the two stones meet.
For Stone A:
Since the initial velocity is zero, we can use the equation:
h = ut + (1/2)at²
Where:
h = height traveled (300m in this case)
u = initial velocity (0 m/s)
a = acceleration (-9.8 m/s²)
t = time
Using the given values, we get:
300m = 0 * t + (1/2) * (-9.8 m/s²) * t²
Simplifying:
300m = -4.9t²
Dividing both sides by -4.9:
t² = -300 / -4.9
t² ≈ 61.22
Finding the square root of both sides:
t ≈ √61.22
t ≈ 7.82 seconds (approx)
For Stone B:
Since the final velocity is zero, we can use the equation:
v = u + at
Where:
v = final velocity (0 m/s)
u = initial velocity (75 m/s)
a = acceleration (-9.8 m/s²)
t = time
Using the given values, we get:
0 = 75 + (-9.8) * t
Simplifying:
-75 = -9.8t
Dividing both sides by -9.8:
t = -75 / -9.8
t ≈ 7.65 seconds (approx)
Since Stone B reaches its highest point at approximately 7.65 seconds, and Stone A takes approximately 7.82 seconds to fall from the tower, we can conclude that the two stones meet around 7.65 seconds after Stone B is projected.
To find where they meet, we can calculate the height at that time by using the equation:
h = ut + (1/2)at²
Substituting the values for Stone B at t = 7.65s:
h = (75 m/s) * (7.65s) + (1/2) * (-9.8 m/s²) * (7.65s)²
Simplifying:
h ≈ 288.62 meters (approx)
Therefore, the two stones meet approximately 7.65 seconds after Stone B is projected, at a height of approximately 288.62 meters above the ground.