To calculate the object's range and maximum height, we need to consider the horizontal and vertical components of its motion separately.

1. Horizontal motion:
The horizontal velocity of the object remains constant throughout its flight. The crosswind speed does not affect the horizontal velocity. Therefore, the range (horizontal distance traveled) can be calculated using the horizontal velocity:

Range = Horizontal velocity * Time

The horizontal velocity is given by: Horizontal velocity = Initial velocity * cos(angle)

where the angle is the launch angle above the horizontal. In this case, the initial velocity is 20 m/s and the launch angle is 30 degrees.

Horizontal velocity = 20 m/s * cos(30°)
Horizontal velocity = 20 m/s * 0.866 (rounded to 3 decimal places)
Horizontal velocity ≈ 17.32 m/s

2. Vertical motion:
The vertical motion of the object is affected by gravity. We can use the equations of motion to calculate the maximum height and time of flight.

a) Maximum height:
The object will reach its maximum height when its vertical velocity becomes zero. We can use the equation:

Vertical velocity = Initial vertical velocity + (acceleration due to gravity * time)

At the maximum height, vertical velocity = 0, so we have:

0 = Vertical velocity + (-9.8 m/s^2) * time

Using the vertical velocity equation:

Vertical velocity = Initial velocity * sin(angle)

Vertical velocity = 20 m/s * sin(30°)
Vertical velocity = 20 m/s * 0.5
Vertical velocity = 10 m/s

Substituting these values into the equation:

0 = 10 m/s + (-9.8 m/s^2) * time
Solving for time, we get:
9.8 m/s^2 * time = 10 m/s
time ≈ 1.02 s

The time of flight is the total time the object spends in the air, which is twice the time to reach maximum height:

Time of flight = 2 * time
Time of flight = 2 * 1.02 s
Time of flight ≈ 2.04 s

Using the time of flight, we can calculate the maximum height using the equation:

Maximum height = Vertical velocity * time - (0.5 * acceleration due to gravity * time^2)
Maximum height = 10 m/s * 1.02 s - (0.5 * 9.8 m/s^2 * (1.02 s)^2)
Maximum height ≈ 10.2 m - 5.021 m
Maximum height ≈ 5.179 m

b) Range:
The range is the horizontal distance traveled by the object in the x-direction. We previously found that the horizontal velocity is approximately 17.32 m/s. The time of flight is 2.04 seconds.

Range = Horizontal velocity * Time
Range = 17.32 m/s * 2.04 s
Range ≈ 35.3808 m

Therefore, the object's range is approximately 35.4 m and its maximum height is approximately 5.18 m.

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