A soccer player kicks a ball into the air at an angle of 36 degrees above the horizontal with a speed of 30m/s

a.How long is the soccer ball in the air?
b. What is the horizontal distance traveled by the ball?
c. What is the maximum height reached by the soccer ball?

a. Until the vertical velocity component has the same sign but opposite drection. T = 2(V sin36)/g

b. (V cos36) * T
c. gH = (Vsin36)^2/2. Solve for H

To answer these questions, we can use the equations of motion for projectile motion. Let's consider the motion of the ball in the horizontal and vertical directions separately.

a. How long is the soccer ball in the air?
The time of flight can be calculated using the equation:
Time of flight = (2 * Initial vertical velocity) / g

Where:
Initial vertical velocity = Initial speed * Sin(θ)
θ = angle of projectile
g = acceleration due to gravity (assumed to be approximately 9.8 m/s^2)

Let's plug in the values:
Initial speed = 30 m/s
θ = 36 degrees
g = 9.8 m/s^2

Initial vertical velocity = 30 m/s * Sin(36 degrees)

Time of flight = (2 * (30 m/s * Sin(36 degrees))) / 9.8 m/s^2

To calculate this, we need to convert the angle to radians:
36 degrees * π / 180 ≈ 0.6283 radians.

The equation becomes:
Time of flight = (2 * (30 m/s * Sin(0.6283))) / 9.8 m/s^2

Using a calculator, we can find that:
Sin(0.6283) ≈ 0.5831

Time of flight = (2 * (30 m/s * 0.5831)) / 9.8 m/s^2
≈ 3.535 seconds

So, the soccer ball is in the air for approximately 3.535 seconds.

b. What is the horizontal distance traveled by the ball?
The horizontal distance traveled can be calculated using the equation:
Horizontal distance = Initial horizontal velocity * Time of flight

The initial horizontal velocity can be calculated using the equation:
Initial horizontal velocity = Initial velocity * Cos(θ)

Let's plug in the values:
Initial velocity = 30 m/s
θ = 36 degrees

Initial horizontal velocity = 30 m/s * Cos(36 degrees)

Using a calculator:
Cos(36 degrees) ≈ 0.8090

Initial horizontal velocity = 30 m/s * 0.8090
≈ 24.27 m/s

Now, let's calculate the horizontal distance:
Horizontal distance = Initial horizontal velocity * Time of flight
= 24.27 m/s * 3.535 seconds
≈ 85.82 meters

So, the horizontal distance traveled by the ball is approximately 85.82 meters.

c. What is the maximum height reached by the soccer ball?
The maximum height can be calculated using the equation:
Maximum height = (Initial vertical velocity)^2 / (2 * g)

Initial vertical velocity can be calculated using the equation:
Initial vertical velocity = Initial velocity * Sin(θ)

Let's plug in the values:
Initial velocity = 30 m/s
θ = 36 degrees
g = 9.8 m/s^2

Initial vertical velocity = 30 m/s * Sin(36 degrees)

Max height = (30 m/s * Sin(36 degrees))^2 / (2 * 9.8 m/s^2)

Using a calculator:
Sin(36 degrees) ≈ 0.5880

Max height = (30 m/s * 0.5880)^2 / (2 * 9.8 m/s^2)
≈ 32.14 meters

So, the maximum height reached by the soccer ball is approximately 32.14 meters.

To determine the answers for these questions, we can break down the motion of the soccer ball into its horizontal and vertical components. Let's calculate each parameter step by step:

a. To find the time the soccer ball stays in the air, we need to calculate the time it takes for the ball to reach its highest point and then return to the ground. We can use the vertical component of the motion to find this time.

We know that the initial vertical velocity (v₀y) is calculated by multiplying the initial velocity (v₀) by the sine of the launch angle (θ), so:
v₀y = v₀ * sin(36°)

Since the ball is launched in the air and returns to the same height, we can use the equation for vertical displacement:
Δy = v₀y * t + (1/2) * (-g) * t²

Since the ball returns to the same height, Δy will be zero. By substituting in the values, we can solve for time (t):

0 = (v₀ * sin(36°)) * t + (1/2) * (-9.8 m/s²) * t²

Simplifying the equation, we have:

(4.9 m/s²) * t² = (v₀ * sin(36°)) * t

which can be further simplified to:

4.9 m/s² * t = v₀ * sin(36°)

Solving for t, we have:

t = (v₀ * sin(36°)) / (4.9 m/s²)

By substituting the given values for velocity and angle, we can calculate the duration the ball remains in the air.

b. To find the horizontal distance traveled by the ball, we need to calculate the horizontal component of the motion. The horizontal velocity (v₀x) remains constant since there is no horizontal force acting on the ball. We can calculate this value using the initial velocity (v₀) and the cosine of the launch angle (θ):

v₀x = v₀ * cos(36°)

Now, we can calculate the horizontal distance (d) traveled by the ball using time (t) obtained from part a:

d = v₀x * t

c. The maximum height reached by the soccer ball can be found by calculating the vertical displacement (Δy) from the starting point. Since the initial vertical velocity at the highest point is zero, the equation to calculate the maximum height is:

Δy = (v₀ * sin(36°)) * t + (1/2) * (-9.8 m/s²) * t²

By substituting the same time obtained from part a, we can calculate the maximum height reached by the ball.

Let's substitute the given values into the equations and calculate the results.