A ball is thrown straight up from the ground
with an initial velocity of 30.4 m/s; at the
same instant, a ball is dropped from the roof
of a building 21.1 m high.
After how long will the balls be at the
same height? The acceleration of gravity is
10 m/s2.
Answer in units of s
d1 + d2 = 21.1 m.
(Vo*t - 0.5g*t^2) + (0.5g*t^2) = 21.1,
The 2nd and 3rd terms cancel:
Vo*t = 21.1,
t = 21.1 / Vo = 21.1 / 30.4 = 0.694 s.
To find out when the two balls will be at the same height, we need to determine the time it takes for each ball to reach that height. Since the ball thrown upward and the ball dropped from the roof have different initial conditions, we'll solve for the time it takes for each ball to reach a height of 21.1 m separately.
For the ball thrown upward, its initial velocity is 30.4 m/s and the acceleration due to gravity is -10 m/s^2 (negative because the velocity is upward). We can use the kinematic equation:
h = v₀t + (1/2)at²
where:
h = height (21.1 m)
v₀ = initial velocity (30.4 m/s)
a = acceleration (-10 m/s^2)
t = time
Plugging in the values, we get:
21.1 = 30.4t + (1/2)(-10)t²
Simplifying the equation, we have:
21.1 = 30.4t - 5t²
Rearranging it into a quadratic equation form:
5t² - 30.4t + 21.1 = 0
We can solve this quadratic equation to find the values of t.
For the ball dropped from the roof, its initial velocity is 0 (as it is simply dropped) and the acceleration due to gravity is also -10 m/s^2. Using the same equation as above, we can solve for the time it takes to reach a height of 21.1 m:
21.1 = 0t + (1/2)(-10)t²
21.1 = -5t²
Dividing both sides by -5, we get:
-4.22 = t²
Since we're looking for the time, we take the square root of both sides:
t = √(-4.22)
However, it's important to note that the square root of a negative number is imaginary. Since time cannot be imaginary, it means that the ball dropped from the roof will never reach the height of 21.1 m (as it's dropped from above that height). Hence, there is no common height for the two balls, and thus, they will not be at the same height.
Therefore, the answer to the question "After how long will the balls be at the same height?" is there is no time at which the balls will be at the same height.