One day I decide I want to see how high I can throw a rock. I throw a rock straight up in the air and it has a speed of 30 m/s when it leaves my hand. I then realize this was foolish, as I need to get out of the way before the rock comes back down and hits me on the head. How many seconds do I have to move out of the way after I throw the rock?
Details and assumptions
The acceleration of gravity is −9.8 m/s^2.
Treat me as a point mass.
Upward motion:
h=v₀t-gt²/2
v= v₀-gt
v=0 => t=v₀/g
h= v²/2g =30²/2•9.8=45.9 m
t=sqrt{2h/g} =sqrt{2•45.9/9.8} = 3.06 s.
t(total)=2•t = 2•3.06 =6.12 s.
To determine how many seconds you have to move out of the way after you throw the rock, we can use the kinematic equations of motion.
First, let's analyze the motion in the vertical direction. Since you throw the rock straight up and it eventually falls back down, its vertical motion can be divided into two parts: the upward motion and the downward motion.
When the rock leaves your hand, it has an initial velocity of 30 m/s in the upward direction. The acceleration due to gravity is acting in the downward direction, and we assume it to be -9.8 m/s^2 since it opposes the upward motion.
The kinematic equation that relates the final velocity (v), initial velocity (u), acceleration (a), and time (t) is:
v = u + at
At the highest point of the rock's trajectory, its vertical velocity becomes 0. Therefore, we can use this equation to find the time it takes for the rock to reach its highest point:
0 = 30 m/s - 9.8 m/s^2 * t (since the final velocity is 0)
Rearranging the equation, we get:
t = 30 m/s / 9.8 m/s^2
t ≈ 3.06 seconds
So, it takes approximately 3.06 seconds for the rock to reach its highest point.
Now, to determine the total time you have to move out of the way, we need to consider the entire trajectory of the rock. Since it takes the same amount of time for the rock to rise to its highest point as it does to fall back down, the total time you have to move out of the way is twice the time it takes for the rock to reach its highest point.
Total time = 2 * t
Total time = 2 * 3.06 seconds
Total time ≈ 6.12 seconds
Therefore, you have approximately 6.12 seconds to move out of the way after throwing the rock.