A hot air balloon is ascending straight up at a constant speed of 6.50 m/s. When the balloon is 13.0 m above the ground, a gun fires a pellet straight up from ground level with an initial speed of 34.0 m/s. Along the paths of the balloon and the pellet, there are two places where each of them has the same altitude at the same time. How far above ground level are these places?
To find the height above ground level where the pellet and the balloon have the same altitude at the same time, we need to consider their respective positions and speeds.
Let's consider the time it takes for the pellet to reach the first point of equal altitude. We can use the formula for the height of an object in free fall:
h = h0 + v0t - (1/2)gt^2
Where:
h is the height of the object at time t,
h0 is the initial height of the object,
v0 is the initial velocity of the object,
g is the acceleration due to gravity (9.8 m/s^2),
t is the time.
For the pellet, since it starts from ground level, h0 = 0. Also, its initial velocity is 34.0 m/s. Let's call the time it takes for the pellet to reach the first point of equal altitude t1.
For the balloon, its initial height is 13.0 m above the ground, and it ascends at a constant speed of 6.50 m/s. The time it takes for the balloon to reach the first point of equal altitude will be the same as t1.
So we can set up the following equation:
0 + 34.0t1 - (1/2)(9.8)t1^2 = 13.0 + 6.50t1
Simplifying this equation gives us a quadratic equation:
4.9t1^2 + 27.5t1 - 13.0 = 0
Now we can solve this quadratic equation to find t1. Using the quadratic formula, we have:
t1 = (-b ± sqrt(b^2 - 4ac)) / (2a)
Where:
a = 4.9
b = 27.5
c = -13.0
Calculating the discriminant (b^2 - 4ac):
discriminant = (27.5)^2 - 4(4.9)(-13.0)
= 756.25 + 254.80
= 1011.05
Since the discriminant is positive, the quadratic equation has two real and distinct solutions.
Using the quadratic formula with the positive square root of the discriminant gives us:
t1 = (-27.5 + sqrt(1011.05)) / (2*4.9)
≈ 2.92 seconds
Now we can substitute this value of t1 back into the equation to find the height above ground level at this time:
h = 13.0 + 6.50t1
= 13.0 + 6.50 * 2.92
≈ 31.99 meters
So, the first point where the balloon and the pellet have the same altitude is approximately 31.99 meters above ground level.
To find the second point of equal altitude, we can repeat the same process but use a different initial height for the pellet. Let's call the time it takes for the pellet to reach the second point of equal altitude t2.
The equation for the pellet becomes:
0 + 34.0t2 - (1/2)(9.8)t2^2 = 0 + 6.50t2
Simplifying this equation gives us:
4.9t2^2 + 27.5t2 = 0
This equation simplifies to:
t2 = (-27.5) / (2*4.9)
≈ -2.82 seconds
The negative value for t2 is not physically meaningful since time cannot be negative. Therefore, we disregard it.
Finally, substitute t2 back into the equation for height to find the second point of equal altitude:
h = 13.0 + 6.50t2
= 13.0 + 6.50 * (-2.82)
≈ -11.53 meters
So, the second point where the balloon and the pellet have the same altitude is approximately 11.53 meters below ground level.