A hot air balloon is ascending with a constant velocity of 5m^s-1 when somebody in the balloon throws a bottle upward with a velocity of 2m^s-1 if the balloon is 55m above tge ground when this happens calculate the time taken for the bottle to reach the ground
h = -1/2 g t^2 + 7 t + 55
h is zero when the bottle hits the ground
0 = -4.9 t^2 + 7 t + 55
you want the positive solution
Yes
To calculate the time taken for the bottle to reach the ground, we can use the equation of motion:
h = ut + (1/2)gt^2
where:
h = height of the balloon above the ground = 55m
u = initial velocity of the bottle = 2m/s (upward)
g = acceleration due to gravity = 9.8m/s^2 (downward)
t = time taken
Since the bottle is thrown upwards, the initial velocity is positive. However, the acceleration due to gravity is in the opposite direction, so it will be negative.
Plugging in the values into the equation, we get:
55 = (2)t + (1/2)(-9.8)t^2
Simplifying the equation further:
0 = (1/2)(-9.8)t^2 + 2t - 55
Now, we have a quadratic equation. To solve for t, we can either use the quadratic formula or factorization. Let's solve it by factoring:
0 = (-4.9)t^2 + 2t - 55
0 = 10t^2 - 4t - 110
0 = 5t^2 - 2t - 55 (dividing by 2 to simplify)
This equation cannot be factored easily, so we'll use the quadratic formula to solve it:
t = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values from the equation, we get:
t = (-(-2) ± √((-2)^2 - 4(5)(-55))) / (2(5))
t = (2 ± √(4 + 1100)) / 10
t = (2 ± √1104) / 10
Now, let's calculate the two possible values for t:
t = (2 + √1104) / 10 ≈ 5.55 seconds (approx.)
t = (2 - √1104) / 10 ≈ -0.55 seconds (discard as time cannot be negative)
Therefore, the time taken for the bottle to reach the ground is approximately 5.55 seconds.
To calculate the time taken for the bottle to reach the ground, we need to consider the motion of the bottle separately from the motion of the balloon.
First, let's calculate the time it takes for the bottle to reach its highest point. We can use the equation of motion:
v = u + at
Where:
v = final velocity of the bottle at its highest point (0 m/s since it will momentarily stop)
u = initial velocity of the bottle (2 m/s)
a = acceleration of the bottle (acceleration due to gravity, -9.8 m/s^2, negative because it is acting in the opposite direction)
t = time taken to reach the highest point
Using the equation, we can rearrange it to solve for time:
t = (v - u) / a
Plugging in the values:
t = (0 - 2) / (-9.8)
t ≈ 0.204 seconds
The time taken for the bottle to reach its highest point is approximately 0.204 seconds.
Now, let's consider the motion of the balloon. Since it is ascending with a constant velocity, the vertical component of its velocity does not change. The bottle's motion is relative to the balloon, so we can combine their velocities to find the time taken for the bottle to hit the ground.
The total time taken for the bottle to reach the ground is equal to the time taken for it to reach its highest point plus the time taken for it to descend from its highest point to the ground.
Since the vertical velocity of the balloon is 5 m/s upward, the total velocity of the bottle relative to the ground is the sum of the velocities: 5 m/s (upward) - 2 m/s (upward) = 3 m/s (upward).
To calculate the time taken for the bottle to reach the ground from its highest point, we can use the same equation of motion:
t = (v - u) / a
Where:
v = final velocity of the bottle when it hits the ground (0 m/s since it will momentarily stop)
u = initial velocity of the bottle (3 m/s)
a = acceleration of the bottle (acceleration due to gravity, -9.8 m/s^2)
t = time taken to reach the ground
Plugging in the values:
t = (0 - 3) / (-9.8)
t ≈ 0.306 seconds
So, the time taken for the bottle to reach the ground is approximately 0.306 seconds.