You throw a ball straight down from an apartment balcony to the ground below. The ball has an initial velocity of 5.20 m/s, directed downward, and it hits the ground 2.10 s after it is released. Find the height of the balcony.
Y = H -5.2t -4.9t^2
is the height above ground at time t.
It hits the ground (Y=0) when t = 2.1 s.
Let Y = 0 and t = 2.10 and solve for H
To find the height of the balcony, we can use the equations of motion under constant acceleration. In this case, the ball is only acted upon by the force of gravity, so the acceleration is equal to the acceleration due to gravity, denoted as "g".
First, we need to find the time it takes for the ball to reach the ground. Since the initial velocity is directed downward, we take the positive direction as downward. The final velocity is zero at the point where the ball hits the ground, so we can use the following equation:
v = u + at
where:
v = final velocity (0 m/s)
u = initial velocity (-5.20 m/s)
a = acceleration (-g)
t = time taken (2.10 s)
Substituting the given values and solving for t, we have:
0 = -5.20 - g(2.10)
Rearranging the equation, we get:
5.20 = 2.10g
Solving for g, we have:
g = 5.20 / 2.10
g ≈ 2.4762 m/s² (approximately)
Now, we can use the equation of motion to find the height of the balcony. The equation can be written as:
s = ut + (1/2)at²
where:
s = distance or height (unknown)
u = initial velocity (-5.20 m/s)
t = time taken (2.10 s)
a = acceleration (-g)
Substituting the known values, we have:
s = -5.20(2.10) + (1/2)(-2.4762)(2.10)²
s ≈ -21.84 + (-2.601) ≈ -24.441
Since we are calculating the height, taking the negative sign into consideration, the height of the balcony is approximately 24.441 meters.