if a person jumps from the tenth flor 30m to a pool thatis 5m away from the building, with what initial horizontally velocity must the person jump
To find the initial horizontal velocity required for the person to jump from the tenth floor and reach a pool 5m away from the building, we can use the principle of projectile motion.
Let's break down the problem into its components:
1. Vertical Motion: Consider the vertical motion of the jump. The vertical distance is 30m, and we can use the equation of motion:
Δy = Viy * t + (1/2) * g * t^2
Where:
- Δy is the vertical displacement (30m),
- Viy is the initial vertical velocity (unknown),
- t is the time of flight, and
- g is the acceleration due to gravity (9.8 m/s^2).
Since the person jumps vertically downward (towards the pool), the initial vertical velocity will be negative (-Viy).
2. Horizontal Motion: Consider the horizontal motion of the jump. The horizontal distance is 5m, and there is no horizontal acceleration. Therefore, the equation of motion is:
Δx = Vix * t
Where:
- Δx is the horizontal displacement (5m),
- Vix is the initial horizontal velocity (unknown), and
- t is the time of flight.
Now, let's solve for the initial horizontal velocity:
1. Vertical Motion:
We can determine the time of flight (t) using the vertical displacement equation:
Δy = Viy * t + (1/2) * g * t^2
Plugging in the known values:
30 = -Viy * t - (1/2) * 9.8 * t^2
Rearranging the equation:
-4.9t^2 - Viy * t + 30 = 0
Solving for t using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -4.9, b = -Viy, and c = 30.
Since the time of flight must be positive, we will use the positive solution.
2. Horizontal Motion:
With the time of flight known, we can solve for the initial horizontal velocity using the horizontal displacement equation:
Δx = Vix * t
Plugging in the known values:
5 = Vix * t
Solving for Vix:
Vix = Δx / t
Substitute the calculated value of t to get the horizontal velocity.
Note: The solution assumes no air resistance and neglects the height of the pool. It is purely a mathematical calculation and may not represent real-world scenarios accurately.