A ball is released from a hot air balloon moving downward with a velocity of
-10.0 meters/second and a height of 1,000 meters. How long did it take the ball to reach the surface of Earth? Given: g = -9.8 meters/second2.
well, the height
y = 1000 - 10.0t - 4.9t^2
just find t when y=0.
To solve this problem, we can use the kinematic equation for the vertical motion of an object:
h = v0t + (1/2)gt^2
where:
h = height (displacement) of the object
v0 = initial velocity of the object
t = time taken
g = acceleration due to gravity
In this case, the ball is released with a downward velocity of -10.0 meters/second and a height of 1,000 meters. We can substitute these values into our equation:
1000 = -10t + (1/2)(-9.8)t^2
Simplifying the equation, we get:
0.5(-9.8)t^2 - 10t + 1000 = 0
Now, we need to solve this quadratic equation for time, t. We can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 0.5(-9.8), b = -10, and c = 1000. Substituting these values into the quadratic formula, we get:
t = (-(-10) ± √((-10)^2 - 4(0.5(-9.8))(1000))) / (2(0.5(-9.8)))
Simplifying further:
t = (10 ± √(100 + 19600)) / (-9.8)
t = (10 ± √19700) / (-9.8)
Now, we can evaluate this equation to find the possible values of t.
To find the time it took for the ball to reach the surface of the Earth, we can use the kinematic equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
In this case, the distance is the height of the balloon, which is 1,000 meters. The initial velocity is -10.0 meters/second (negative because it is moving downward) and the acceleration is -9.8 meters/second^2 (negative because it is in the opposite direction of the velocity).
Substituting these values into the equation, we get:
1000 = -10 * t + (1/2) * (-9.8) * t^2
Simplifying the equation, we have:
1000 = -10t - 4.9t^2
Rearranging the equation, we get a quadratic equation:
4.9t^2 + 10t - 1000 = 0
Now we can solve this quadratic equation using methods like factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = 4.9, b = 10, and c = -1000. Substituting these values into the quadratic formula, we get:
t = (-(10) ± sqrt((10)^2 - 4 * 4.9 * (-1000))) / (2 * 4.9)
t = (-10 ± sqrt(100 + 19600)) / 9.8
t = (-10 ± sqrt(19700)) / 9.8
t ≈ (-10 ± 140.35) / 9.8
So we have two solutions for t:
t ≈ (130.35) / 9.8 ≈ 13.3 seconds
t ≈ (-150.35) / 9.8 ≈ -15.3 seconds
Since time cannot be negative in this context, we discard the negative solution. Therefore, the ball took approximately 13.3 seconds to reach the surface of the Earth.