A ball with a mass 0.15kg is thrown upward with an initial velocity 20m/s from the roof of a building 30m high. Neglect air resistance.
Find the max height above the ground that the ball reaches.
Assuming the ball misses the building find the time that it hits the ground
the mass of the ball does not matter.
The height as a function of time is
h(t) = 30 + 20t - 4.9t^2
so, just find the vertex of the parabola, and solve for t when h=0. Just good old Algebra I.
so this is for a diff eq class. how did you use differential equations to come up with that height equation?
you have constant acceleration
h" = -9.8 m/s^2
so, the speed is h' = -9.8t+Vo m/s
Vo=20, so
h' = -9.8t+20
h= -4.9t^2+20t+Ho
Ho=20, so
h = -4.9t^2+20t+30
To find the maximum height above the ground that the ball reaches, we can use the following steps:
1. Determine the time it takes for the ball to reach its maximum height.
2. Use this time to find the maximum height.
Step 1: Determine the time to reach maximum height
We can use the kinematic equation to find the time it takes for the ball to reach its maximum height:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s at the maximum height)
u = initial velocity (20 m/s)
a = acceleration (which is equal to the acceleration due to gravity, -9.8 m/s^2)
s = displacement (unknown)
Rearranging the equation, we get:
0 = 20^2 + 2(-9.8)s
Simplifying further:
0 = 400 - 19.6s
Solving for s, we find:
s = 400 / 19.6
s ≈ 20.41 m
This means it takes approximately 20.41 meters for the ball to reach its maximum height.
Step 2: Use the time to find the maximum height
To find the maximum height, we can use the following equation:
s = ut + (1/2)gt^2
Where:
s = displacement (the maximum height above the ground, unknown)
u = initial velocity (20 m/s)
t = time taken to reach maximum height (unknown)
g = acceleration due to gravity (-9.8 m/s^2)
Since we have already found the time to reach maximum height (20.41 m), we can substitute it into the equation:
s = 20(20.41) + (1/2)(-9.8)(20.41)^2
Simplifying further:
s ≈ 205.77 m
Therefore, the ball reaches a maximum height of approximately 205.77 meters above the ground.