a stone is thrown straight up from a 50m high building and hits the ground at a speed of 37.1m/s. What is the initial velocity and what is the maximum height above the ground that the stone reaches?
For maximum height, equate PE and KE
mgh=(1/2)mv²
h=(1/2)37.1²/9.8=70.2 m
at 50m, KE is reduced by mgh
=>
(1/2)mv²=(1/2)m(37.1)²-mg(50)
v=sqrt((37.1²-2(g)(50))
=19.9 m/s
To calculate the initial velocity and the maximum height reached by the stone, we can make use of the basic equations of motion.
Step 1: Find the initial velocity:
When the stone is thrown straight up, its final velocity (when it reaches the ground) will be equal in magnitude but opposite in direction to its initial velocity. Therefore, the value of the final velocity can be considered as -37.1 m/s (negative indicates opposite direction).
We can use the equation for final velocity, initial velocity, acceleration, and time:
v = u + at
Here,
v = final velocity = -37.1 m/s
u = initial velocity (unknown)
a = acceleration due to gravity (approximately -9.8 m/s^2, negative because the stone is moving upward)
t = time taken to reach the ground (unknown)
Since the stone is thrown up and comes back to the same height, the time taken to reach the ground will be equal to the time taken to reach the maximum height. Let's denote this time as t_max.
So, the equation becomes:
-37.1 = u - 9.8 * t_max
Step 2: Find the maximum height:
To find the maximum height reached, we can use the equation for displacement:
s = ut + (1/2) * a * t^2
At the maximum height, the final velocity will be zero (as the stone momentarily stops before falling back down), so we can plug in the values:
s_max = 0 + (1/2) * (-9.8) * t_max^2
Here,
s_max = maximum height reached (unknown)
u = initial velocity (from step 1)
a = acceleration due to gravity = -9.8 m/s^2
t_max = time taken to reach maximum height (unknown)
Now we have two equations:
-37.1 = u - 9.8 * t_max
s_max = (1/2) * (-9.8) * t_max^2
Solving these two equations simultaneously will give us the values for the initial velocity (u) and the maximum height (s_max).
Please note that we need more information, specifically the value of t_max or s_max, to calculate the exact values.