The university skydiving club has asked you to plan a stunt for an air show. In this stunt, two skydivers will step out of opposite sides of a stationary hot air balloon 2,500m above the ground. the second skydiver will leave the ballon 20 seconds after the first skydiver but you want them both to land on the ground at the same time. the show is planned for a day with no wind so assume all motion is vertical. to ger rough idea of the situation, assume that a skydiver will fall with a constant acceleration of 9.8m/s^2 before the parachute opens. As soon as the parachute is opened, the skydiver falls with a constant velocity of 3.2m/s. if the first skydiver waits 3s after stepping out of the balloon before opening her parachute, How long must the second skydiver wait after leaving the balloon before opening his parachute?
How to solve for time?
To solve for the time the second skydiver must wait after leaving the balloon before opening the parachute, we need to consider the different phases of their descent.
1. Calculate the time it takes for the first skydiver to reach the ground:
- Distance fallen by the first skydiver = 2,500m
- Acceleration due to gravity = 9.8 m/s^2 (constant acceleration before parachute opens)
- Initial velocity of the first skydiver = 0 m/s (since they start from rest)
- Using the equation: distance = (initial velocity * time) + (0.5 * acceleration * time^2)
- Plug in the values: 2,500m = (0 * t) + (0.5 * 9.8 m/s^2 * t^2)
- Solve for time: 2,500m = 4.9 m/s^2 * t^2
- Rearrange the equation: t^2 = 2,500m / 4.9 m/s^2
- Calculate: t = sqrt(2,500m / 4.9 m/s^2) = 31.6 seconds (approx.)
2. Calculate the time it takes for the second skydiver to reach the ground:
- The second skydiver waits for 20 seconds after the first skydiver before jumping.
- Therefore, the second skydiver will have 20 seconds less time to reach the ground compared to the first skydiver.
- Subtracting 20 seconds from the calculated time for the first skydiver: 31.6 seconds - 20 seconds = 11.6 seconds
Therefore, the second skydiver must wait approximately 11.6 seconds after leaving the balloon before opening his parachute in order to land at the same time as the first skydiver.
To solve for the time the second skydiver must wait after leaving the balloon before opening his parachute, we can use the equations of motion for free fall. The key is to find the time it takes for both skydivers to reach the ground.
Let's break down the problem step by step:
1. The first skydiver is already given to wait 3 seconds before opening her parachute. This means she will experience free fall for 3 seconds before reaching a constant velocity of 3.2 m/s due to the parachute.
2. Let's calculate the time it takes for the first skydiver to reach the ground. We can use the equation of motion: s = ut + (1/2)at^2, where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.
The skydiver starts at a height of 2500 meters and falls under the acceleration due to gravity (9.8 m/s^2) for 3 seconds. Therefore, the distance fallen during free fall is given by: s = (0 m/s)(3 s) + (1/2)(9.8 m/s^2)(3 s)^2 = 44.1 m.
After opening the parachute, the skydiver falls with a constant velocity of 3.2 m/s. Therefore, the remaining distance to the ground is: s = (3.2 m/s)(t), where t is the time the skydiver spends in constant velocity.
Thus, the total distance fallen by the first skydiver is: 44.1 m + 3.2 m/s * t.
3. Now, let's find the time it takes for the second skydiver to reach the ground, considering that he opens his parachute 20 seconds after the first skydiver.
The second skydiver starts at the same height of 2500 meters, but he waits an additional 20 seconds before opening his parachute. This means he experiences free fall for 3 + 20 = 23 seconds before reaching a constant velocity of 3.2 m/s.
Using the same equation of motion as before, the distance fallen during free fall is given by: s = (0 m/s)(23 s) + (1/2)(9.8 m/s^2)(23 s)^2 = 2562.7 m.
After opening the parachute, the skydiver falls with a constant velocity of 3.2 m/s. Therefore, the remaining distance to the ground is: s = (3.2 m/s)(t), where t is the time the skydiver spends in constant velocity.
Thus, the total distance fallen by the second skydiver is: 2562.7 m + 3.2 m/s * t.
4. Since both skydivers must land on the ground at the same time, the total distance fallen by the first skydiver must be equal to the total distance fallen by the second skydiver.
Setting the two distances equal to each other: 44.1 m + 3.2 m/s * t = 2562.7 m + 3.2 m/s * t.
5. We can now solve for the time t by subtracting 44.1 m from both sides of the equation: 3.2 m/s * t = 2562.7 m - 44.1 m.
Simplifying: 3.2 m/s * t = 2518.6 m.
6. Finally, to solve for t, divide both sides of the equation by 3.2 m/s: t = 2518.6 m / 3.2 m/s.
Evaluating the right side of the equation: t ≈ 787.06 seconds.
Therefore, the second skydiver must wait approximately 787.06 seconds (or about 13 minutes and 7 seconds) after leaving the balloon before opening his parachute in order to land at the same time as the first skydiver.
The second parachutist must open the parachute at the moment that she catches up with the first, i.e. has fallen the same distance as the first.
1. Establish the time/distance relation of first parachutist.
For the first 3 seconds,
distance = 0 + 9.8*(3²) = 44.1 m
total distance at time t (t≥3 s)
=44.1+3.2(t-3)
=3.2t+34.5
2. Establish the time/distance relation of second parachutist before her parachute was open (for t≥20):
distance=(1/2)9.8((t-20)²)
=4.9t²-196t+1960
Equate times and solve for time the two parachutists were at the same elevation
(note that time is measured from the first jump)
32t+345=49t²-1960t+19600
and solve for t.
Do subtract 20 seconds from t since the question asks for time since leaving the balloon. Reject all t<20 seconds.