An object is dropped from a balloon from a height of 500m. (a) How long does it take to reach the ground?
(b) If the object has an initial velocity of 12m/s, how long does it take to reach the ground? How far will it fall in 5 secnds?
the distance
s = 1/2 at^2
So,
500 = 4.9t^2
Now you have the time.
You need to say what the velocity's direction is. As you recall,
s = vt + 1/2 at^2
It makes a big difference whether v is positive or negative.
In 5 seconds it will fall 4.9*5^2 meters.
To answer these questions, we can use the equations of motion for free fall. The key equation we will be using is:
d = vit + 1/2at^2
where:
d = distance traveled
vi = initial velocity
a = acceleration
t = time
Let's break down the questions one by one:
(a) How long does it take to reach the ground?
To find the time it takes for the object to reach the ground, we can use the equation of motion, with d = 500m, vi = 0 (since it is dropped), and a = 9.8 m/s^2 (acceleration due to gravity). Plugging these values into the equation, we get:
500 = 0 * t + 1/2 * 9.8 * t^2
Simplifying the equation:
500 = 4.9t^2
Divide both sides of the equation by 4.9:
102.04 = t^2
Take the square root of both sides:
t ≈ 10.1 seconds
Therefore, it takes approximately 10.1 seconds for the object to reach the ground when dropped from a height of 500m.
(b) If the object has an initial velocity of 12m/s, how long does it take to reach the ground? How far will it fall in 5 seconds?
In this case, we need to calculate the time it takes for the object to reach the ground with an initial velocity of 12m/s and find the distance it falls in 5 seconds.
To find the time it takes to reach the ground, we use the same equation of motion as before, but with vi = 12m/s:
500 = 12 * t + 1/2 * 9.8 * t^2
Simplifying the equation, we get:
4.9t^2 + 12t - 500 = 0
This is a quadratic equation. We can solve it using the quadratic formula or factoring. By factoring, we find:
(4.9t - 50)(t + 10) = 0
Setting each factor equal to zero and solving for t, we get:
4.9t - 50 = 0 --> t = 10.2 seconds (ignoring the negative solution)
Therefore, it takes approximately 10.2 seconds for the object to reach the ground when dropped with an initial velocity of 12m/s.
To find the distance it falls in 5 seconds, we use the equation of motion with vi = 12m/s:
d = 12 * 5 + 1/2 * 9.8 * 5^2
d = 60 + 1/2 * 9.8 * 25
d = 60 + 122.5
d ≈ 182.5 meters
Therefore, the object falls approximately 182.5 meters in 5 seconds.