A hoodlum throws a stone vertically downward with an initial speed v0 from the roof of a building, a height h above the ground. (a) How long does it take the stone to reach the ground? (b) What is the speed of the stone at impact? Give your answers in terms of the given variables and g.
To determine the time it takes for the stone to reach the ground, we can use the equations of motion for an object in free fall. The free fall motion can be described using the equation:
h = v0*t + (1/2)*g*t^2
where:
h = height of the building
v0 = initial velocity of the stone (downward direction)
g = acceleration due to gravity
t = time taken for the stone to reach the ground
Since the stone is thrown vertically downward, the initial velocity, v0, is negative.
(a) To find the time it takes for the stone to reach the ground, we set h = 0 in the above equation:
0 = v0*t + (1/2)*g*t^2
Since the stone is thrown downward, the initial velocity, v0, is negative. Let's assume it as -v0 for simplicity.
0 = (-v0)*t + (1/2)*g*t^2
Rearranging this equation, we get:
(1/2)*g*t^2 = v0*t
Dividing both sides of the equation by t and multiplying by 2, we have:
gt = 2v0
Now, solving for t, we get:
t = 2v0/g
(b) To find the speed of the stone at impact, we can use the equation for velocity in free fall:
v = v0 + gt
Substituting the value of v0 as -v0, we get:
v = -v0 + gt
Since the stone reaches the ground, the height is zero. Therefore, the final velocity, v, can also be considered as the speed of the stone at impact.
So, the speed of the stone at impact is -v0 + gt.
Remember, for both parts (a) and (b), the answers are in terms of the given variables h, v0, and g.
2.5s
24m/s