A stone is thrown straight up. To reach a height of 6.74 m above its release point, what must its initial speed be in m/s
To determine the initial speed of the stone, we can use the principles of projectile motion. When the stone is thrown straight up, it decelerates until it reaches its highest point and then falls back down.
At the highest point of its trajectory, the stone's vertical velocity will be zero. Using the equation of motion:
Vf = Vi + at
where Vf is the final velocity, Vi is the initial velocity, a is the acceleration, and t is the time, we can determine the initial velocity.
Since the stone is thrown straight up against the force of gravity, its acceleration is equal to the acceleration due to gravity, denoted as g, which is approximately 9.8 m/s² (neglecting air resistance).
At its highest point, the stone will have traveled a vertical distance equal to the desired height of 6.74 m above its release point. At this point, the final velocity is zero. Plugging these values into the equation, we have:
0 = Vi + (-9.8 m/s²) * t
where Vi is the initial velocity and t is the time it takes to reach the highest point.
We can rearrange the equation to solve for t:
t = Vi / 9.8
Now we need to calculate the time it takes for the stone to reach its highest point. The stone's initial velocity is the same as its final velocity just before it starts descending. We can determine the time it takes to reach this point by dividing the total time of the entire motion by 2.
The total time of the motion can be found using the following equation:
h = Vi * t + (1/2) * (-9.8 m/s²) * t²
where h is the total vertical displacement (6.74 m) and t is the time to reach the highest point.
Plugging in the known values:
6.74 m = Vi * (t/2) + (1/2) * (-9.8 m/s²) * (t/2)²
Now, we can solve this equation to find the value of t:
6.74 m = (Vi/2) * t - (4.9 m/s²) * (t/2)²
Next, we'll substitute the value of t in the equation with t = Vi / 9.8:
6.74 m = (Vi/2) * (Vi / 9.8) - (4.9 m/s²) * ((Vi / 9.8)/2)²
Simplifying this equation will provide the value of Vi, which is the initial velocity of the stone.