A soccer game is played. A goal keepers kicks the ball 88m . If the ball had an elevation of 40 degree angle upon its departure calculate : a) the initial velocity of the ball// b) the max. height of the ball// c) the time of flight of the ball
To calculate the initial velocity, max height, and time of flight of the ball, we can use some basic principles of projectile motion.
a) Calculating the initial velocity:
To calculate the initial velocity of the ball, we can break the initial velocity into its horizontal and vertical components. The horizontal component of the velocity remains constant throughout the motion, while the vertical component is influenced by gravity.
The horizontal component of the velocity (Vx) remains constant and can be found using the formula:
Vx = V × cos(θ)
where V is the initial velocity of the ball and θ is the launch angle (40 degrees). Since no horizontal force affects the projectile, Vx does not change during the motion.
The vertical component of the velocity (Vy) can be found using the formula:
Vy = V × sin(θ)
where V is the initial velocity of the ball and θ is the launch angle (40 degrees).
To find the initial velocity of the ball (V), we can use Vy value:
V = Vy / sin(θ)
V = (88m) / sin(40°)
b) Calculating the maximum height:
To calculate the maximum height of the ball, we can use the formula for vertical displacement:
Δy = (Vy²) / (2g)
where Δy is the vertical displacement, Vy is the vertical component of velocity, and g is the acceleration due to gravity (9.8 m/s²).
The maximum height reached by the ball will occur when the vertical velocity becomes zero. Therefore, the maximum height is equal to the vertical displacement:
Max height = Δy = (Vy²) / (2g)
c) Calculating the time of flight:
The time of flight of the ball is the total time it takes for the ball to reach the ground. To calculate this, we can use the formula for time:
t = (2 × Vy) / g
Using the values calculated for Vy, g, and the formulas above, you can find the answers to parts a, b, and c of the question.
To calculate these values, we can use the kinematic equations of motion. We'll start by splitting the initial velocity of the ball into its horizontal and vertical components.
a) Calculating the initial velocity:
Given:
- Distance traveled by the ball, d = 88 m
- Launch angle, θ = 40°
To calculate the initial velocity (vi), we can use the following equation:
vi = d / (cos θ)
Substituting the values, we get:
vi = 88 m / (cos 40°)
Using a calculator, we find:
vi ≈ 107.64 m/s
b) Calculating the maximum height:
The maximum height can be determined using the following equation:
h = (vi² * sin² θ) / (2 * g)
Where g is the acceleration due to gravity, approximately 9.8 m/s².
Substituting the values, we have:
h = (107.64 m/s)² * sin² 40° / (2 * 9.8 m/s²)
Calculating this expression, we find:
h ≈ 51.20 m
c) Calculating the time of flight:
To find the time of flight (t), we can use the equation:
t = (2 * vi * sin θ) / g
Substituting the values, we get:
t = (2 * 107.64 m/s * sin 40°) / 9.8 m/s²
Calculating this expression, we find:
t ≈ 10.99 s
Therefore, the initial velocity of the ball is approximately 107.64 m/s, the maximum height is approximately 51.20 m, and the time of flight is approximately 10.99 s.