Question

a hot air balloon is rising straight up at a constant speed of 7 m/s. when the balloon is 12.0 m/s above the ground, a gun fires a pellet straight up from ground level with an initial speed of 30.0 m/s. Along the paths of the ballon and pellet, there are two places where each of then has the same altitude at the same time. How far above ground are these places?

Answer 1

To find the points where the hot air balloon and the pellet have the same altitude at the same time, we analyze their respective vertical motions. By using the equations of motion, we can determine when the pellet reaches the same altitude as the balloon and find the altitude above the ground at that time.

Step-by-step explanation:

To find the points where the hot air balloon and the pellet have the same altitude at the same time, we need to analyze their respective vertical motions. The hot air balloon is rising straight up at a constant speed, while the pellet is fired straight up from the ground.

First, let's determine the time it takes for the pellet to reach the same altitude as the balloon. We can use the equations of motion to do this. The initial velocity of the pellet is 30.0 m/s, and we can assume its acceleration is -9.8 m/s² (negative due to the direction). Since the initial position is 0 m and the final position is 12.0 m (same altitude as the balloon), we can use the kinematic equation:

d = v0t + 0.5at²

Substituting the given values, we have:

12.0 = 30.0t + 0.5(-9.8)t²

Simplifying and rearranging the equation gives us:

4.9t² - 30t + 12.0 = 0

We can solve this quadratic equation to find the value of t, which represents the time it takes for the pellet to reach the same altitude as the balloon. Once we have t, we can substitute it back into the equation to find the altitude above the ground at that time.

Now, let's examine the hot air balloon. Since it is rising straight up at a constant speed of 7 m/s, we can calculate the time it takes for the balloon to reach the same altitude as the pellet. We can use the equation:

altitude = initial altitude + velocity × time

Substituting the given values:

12.0 = 0 + 7t

By rearranging the equation, we find:

t = 12.0 / 7

Once we have the value of t, we can substitute it back into the equation to find the altitude above the ground at that time.

Answer 2

For the balloon,
Y = 7 t + 12
where t is the time the pellet is fired.
For the pellet,
Y = 30 t - (g/2) t^2
= 30 t - 4.9 t^2


Solve for the two t's when the Y's are equal.
4.9 t^2 - 23 t +12 = 0

gravity is 9.8