A ball is thrown upward from the ground with an initial speed of 26.2 m/s; at the same instant a ball is dropped from rest from a building 14 m high. After how long will the balls be at the same height?
d1 + d2 = 14 m,
26.2 m/s * t + 0.5 * 9.8 m/s^2 *t^2 = 14 m,
26.2t + 4.9t^2 = 14,
4.9t^2 + 26.2t - 14 = 0,
Use quadratic formula to solve for t:
t = (-26.2 +- sqrt(686.44 + 274.4)) /9.8,
t = (-26.2 +- 31) / 9.8
t = 0.49 s or -5.84 s, select + t:
t = 0.49 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.
Let's start by considering the ball that was thrown upward from the ground. The initial velocity of this ball is 26.2 m/s, and since it's thrown upward, we can assume the acceleration due to gravity acts in the opposite direction, so its value is -9.8 m/s². We know that the initial position is 0 (since we are measuring from the ground), and we want to find the time when it reaches the same height as the ball dropped from the building.
Using the kinematic equation for displacement:
s = ut + (1/2)at²
where:
s = displacement
u = initial velocity
t = time
a = acceleration
For the ball thrown upward, s is the height reached by the ball. And since the ball was thrown upward, the final velocity is 0 when it reaches the maximum height.
So, we have:
s₁ = 0 + (1/2)(-9.8)t₁²
Now let's consider the ball dropped from rest from a building 14 m high. In this case, the initial velocity is 0, and the acceleration due to gravity acts in the downward direction, so its value is +9.8 m/s². The displacement is the height of the building, which is 14 m.
Using the same kinematic equation as before, we have:
s₂ = 0 + (1/2)(9.8)t₂²
For both cases, s₁ and s₂ represent the same height. So we can set them equal to each other:
0 + (1/2)(-9.8)t₁² = 0 + (1/2)(9.8)t₂²
Simplifying the equation:
(-4.9)t₁² = (4.9)t₂²
Dividing both sides by 4.9:
t₁² = t₂²
Now we can take the square root of both sides to solve for t:
t₁ = t₂
Therefore, the time it takes for both balls to be at the same height is the same.