In 1993, Javier Sotomayor set a world record of 2.43 m in the men's outdoor high jump. He is 193 cm(6ft 4in) tall. BY treating his body as a point located at half his height, given that he left the ground a horizontal distance from the bar 1.5 m at a takeoff angle of 65 degrees, determine Javier takeoff speed?
To determine Javier's takeoff speed, we can use the principles of projectile motion. Let's break down the problem into different components and calculate the necessary values:
1. Initial data:
- Time of flight (t)
- Horizontal distance (d) = 1.5 m
- Takeoff angle (θ) = 65 degrees
2. Acceleration due to gravity:
- We know that the acceleration due to gravity (g) is approximately 9.8 m/s². This value remains constant throughout the motion.
3. Horizontal motion:
- During the horizontal motion, there is no acceleration acting in this direction. Therefore, the horizontal component of velocity (Vx) remains constant.
- The horizontal distance (d) is given as 1.5 m.
4. Vertical motion:
- During the vertical motion, gravity is the only force acting. Therefore, the vertical component of velocity (Vy) changes during the motion.
- At the highest point of the jump, the vertical component of velocity (Vy) becomes zero.
Now, let's calculate the necessary values step by step:
Step 1: Calculate the time of flight (t):
- We can use the formula: t = 2 * Vy_max / g.
- Since Vy_max = 0 at the highest point of the jump, the time of flight (t) will be the same for both upward and downward motions.
- Therefore, we can use the given height of the jump to calculate t using the formula: h = V₀y * t - 0.5 * g * t², where h = 2.43 m.
- Solving this equation, we find t.
Step 2: Calculate the initial vertical component of velocity (V₀y):
- We can use the formula: V₀y = Vy_max + g * t / 2.
- Since Vy_max = 0 at the highest point of the jump, we can find V₀y using the calculated time of flight (t).
Step 3: Calculate the initial horizontal component of velocity (V₀x):
- We can use the formula: V₀x = d / t.
- Here, d is the horizontal distance given as 1.5 m, and t is the calculated time of flight.
Step 4: Calculate the initial takeoff speed (V₀):
- We can use the formula: V₀ = √(V₀x² + V₀y²).
- Using the calculated initial horizontal component of velocity (V₀x) and initial vertical component of velocity (V₀y), we can find V₀.
By following these steps and plugging in the given values, you can calculate Javier Sotomayor's takeoff speed.
Range = Vo^2*sin(2A)/g = 1.5 m.
Vo^2*sin(130)/9.8 = 1.5
0.078168Vo^2 = 1.5
Vo^2 = 19.19
Vo = 4.38 m/s.