a ball is thrown vertically upward from the ground level with an initial velocity "Vo" - V sub zero
a. derive a formula for the time at w/c the ball hit the ground
b. use the formula to prove that the impact velocity by the ball is -Vo
h(t) = Vot - 1/2 a t^2
= t(Vo - a/2 t)
h=0 when t=0 or t=2Vo/a)
v(t) = Vo - at
at t=2Vo/a, v = Vo - a(2Vo/a) = -Vo
Makes sense, since the ball falls just as far as it rises.
a. To derive a formula for the time at which the ball hits the ground, we need to consider the motion of the ball. When the ball is thrown upward, the only force acting on it is gravity, which causes it to decelerate.
Let's assume the upward direction is positive. The initial velocity of the ball, denoted as Vo, is positive since it is being thrown upward. The acceleration due to gravity, denoted as g, is negative since it acts downward.
Using the equation of motion:
v = u + at,
where
v = final velocity (which is 0 m/s when the ball hits the ground),
u = initial velocity (Vo),
a = acceleration (-g),
t = time.
Substituting the given values, we get:
0 = Vo - gt.
Solving for t, we have:
gt = Vo,
t = Vo / g.
Therefore, the formula for the time at which the ball hits the ground is t = Vo / g.
b. To prove that the impact velocity of the ball is -Vo, we can use the formula for velocity:
v = u + at,
where
v = final velocity,
u = initial velocity,
a = acceleration,
t = time.
In this case, when the ball hits the ground, the final velocity v is -Vo (since it is moving downward), the initial velocity u is Vo, the acceleration a is -g, and the time t is the one we found previously as t = Vo / g.
Substituting these values into the equation, we have:
-Vo = Vo + (-g) * (Vo / g).
Simplifying,
-Vo = Vo - Vo,
-Vo = -Vo.
Therefore, the impact velocity of the ball is indeed -Vo.