A place kicker must kick a football from a point 36 m from the goal, and the ball must clear the crossbar, which is 3.05m high. When kicked, the ball leaves the ground with a speed of 20.0m/s at an angle of 53 degrees to the horizontal.
a. by how much does the ball clear or fall short of clearing the crossbar?
b. does the ball approach the crossbar while still rising or while falling?
how do i do thiss??? thanks:)
Calculate how long it takes for the ball to get to the goal post. Then calculate its vertical velocity component and elevation at that time. Compare the vertical position to 3.05 m to see if it cleared the bar. The sign of the vertical velocity component at that time will tell your the answer to b.
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To solve this problem, we can break it down into two components: the horizontal and vertical motion of the ball. Let's first calculate the time it takes for the ball to reach the crossbar.
1. Horizontal Motion:
The horizontal velocity of the ball remains constant throughout the motion. Therefore, we can calculate the time it takes for the ball to reach the crossbar using the horizontal distance traveled and the horizontal velocity.
Distance (d) = 36 m
Horizontal Velocity (Vx) = velocity * cos(angle) = 20.0 m/s * cos(53°)
Time (t) = d / Vx
2. Vertical Motion:
The vertical motion of the ball is affected by gravity. We can calculate the maximum height the ball reaches and determine if it is greater than the crossbar's height.
Initial Vertical Velocity (Vy) = velocity * sin(angle) = 20.0 m/s * sin(53°)
Maximum Height (H) = (Vy^2) / (2 * g)
where g is the acceleration due to gravity (approximated as 9.8 m/s²)
Now, let's calculate the values:
1. Horizontal Motion:
Vx = 20.0 m/s * cos(53°) = 20.0 m/s * 0.6018 ≈ 12.036 m/s
t = 36 m / 12.036 m/s ≈ 2.99 s
2. Vertical Motion:
Vy = 20.0 m/s * sin(53°) = 20.0 m/s * 0.7986 ≈ 15.972 m/s
H = (15.972 m/s)^2 / (2 * 9.8 m/s²) ≈ 12.91 m
Now, let's answer the questions:
a. By how much does the ball clear or fall short of clearing the crossbar?
The ball reaches a maximum height of approximately 12.91 m, which is greater than the crossbar's height of 3.05 m. Therefore, the ball clears the crossbar by:
Clearance = Maximum Height - Crossbar Height
Clearance = 12.91 m - 3.05 m = 9.86 m
b. Does the ball approach the crossbar while still rising or while falling?
Since the ball clears the crossbar, it means that it approaches the crossbar while still rising, as the maximum height occurs before reaching the crossbar.
So, the ball clears the crossbar by approximately 9.86 m and approaches it while still rising.