A ball is thrown downward from a balcony that is 19.6m above the ground one kid throws downward at 14.7m/s, at the same time another student throws a ball upward from the balcony at the same speed. what is the difference of the time the balls are in the air and what is the velocity when the balls hit the ground?
To find the time difference and the velocity when the balls hit the ground, we can use the kinematic equations of motion.
For the ball thrown downward, we will consider the positive direction as downwards, so the initial velocity (u) is +14.7 m/s (downwards), the distance (h) is 19.6 m (height of the balcony above the ground), and the acceleration (a) is +9.8 m/s² (due to gravity acting downwards).
1. Finding the time for the ball thrown downward:
We can use the equation: h = ut + (1/2)at², where h is the distance, u is the initial velocity, a is the acceleration, and t is the time.
Substituting the given values:
19.6 = (14.7)t + (1/2)(9.8)(t²)
Simplifying the equation:
19.6 = 14.7t + 4.9t²
This is a quadratic equation. Rearranging it to standard form:
4.9t² + 14.7t - 19.6 = 0
Solving this equation using the quadratic formula, we find:
t = (-14.7 ± √(14.7² - 4(4.9)(-19.6))) / (2(4.9))
Calculating the values, we get:
t ≈ 1.285 s (ignoring the negative solution)
2. Finding the time for the ball thrown upward:
Since the ball thrown upward has the same initial speed, the initial velocity (u) is -14.7 m/s (upwards).
Using the same formula: h = ut + (1/2)at²
19.6 = (-14.7)t + (1/2)(-9.8)(t²)
Rearranging and simplifying:
4.9t² - 14.7t - 19.6 = 0
Solving for t again using the quadratic formula:
t ≈ 2.835 s (ignoring the negative solution)
3. Finding the time difference:
The time difference between the balls is:
2.835 - 1.285 ≈ 1.55 s
4. Finding the velocity when both balls hit the ground:
The final velocity (v) when an object hits the ground after falling from a height can be found using the formula: v = u + at, where u is the initial velocity, a is the acceleration, and t is the time.
For the ball thrown downward:
v = 14.7 + (9.8)(1.285)
v ≈ 27.66 m/s (downwards)
For the ball thrown upward:
v = -14.7 + (9.8)(2.835)
v ≈ -12.37 m/s (upwards)
Note: The negative sign indicates the direction of velocity. In this case, downwards is considered positive and upwards is considered negative.