a place kicker kick a ball 36m from a goal .which is 3.05 m high.it leaves the ground 20 m/s at an angle 53 to the horizontal.how much the ball clear or fall short of clearing the bar
Use d = d0 + v0t + (1/2)at^2
In the x direction:
x = x0 + v0t + (1/2)at^2
solve for t
Plug t into:
y = y0 + v0t + (1/2)at^2
y - 3.05 = The distance by which the ball will clear, or fall short of, the bar.
Velocity in x direction = v0cos(53)
Velocity in y direction = v0sin(53)
To find out how much the ball clears or falls short of clearing the bar, we need to analyze the vertical component of the ball's motion.
First, we need to break down the initial velocity into its horizontal and vertical components using trigonometry. We can find the vertical component using the equation:
Vertical Component (Vy) = Initial Velocity (v) * sin(theta)
Given that the initial velocity is 20 m/s and the angle is 53 degrees, we can calculate the vertical component:
Vertical Component (Vy) = 20 m/s * sin(53 degrees) ≈ 16.04 m/s
Now, we can use this vertical component to calculate the time of flight (t) for the ball to reach its maximum height. We can use the equation:
Time of Flight (t) = (2 * Vertical Component (Vy)) / Acceleration due to gravity (g)
The acceleration due to gravity near the Earth's surface is approximately 9.8 m/s^2. Hence, calculating the time of flight:
Time of Flight (t) = (2 * 16.04 m/s) / 9.8 m/s^2 ≈ 3.28 seconds
Now, using this time of flight, we can calculate the upward distance traveled by the ball at its maximum height using the equation:
Upward Distance (h) = (Vertical Component (Vy))^2 / (2 * Acceleration due to gravity (g))
Plugging in the values:
Upward Distance (h) = (16.04 m/s)^2 / (2 * 9.8 m/s^2) ≈ 13.07 m
Next, we need to calculate the horizontal distance (x) traveled by the ball by using the equation:
Horizontal Distance (x) = Horizontal Component (Vx) * Time of Flight (t)
To find the horizontal component, we can use the equation:
Horizontal Component (Vx) = Initial Velocity (v) * cos(theta)
Calculating the horizontal component:
Horizontal Component (Vx) = 20 m/s * cos(53 degrees) ≈ 11.65 m/s
Now, we can calculate the horizontal distance by plugging in the values:
Horizontal Distance (x) = 11.65 m/s * 3.28 s ≈ 38.21 m
Finally, to determine how much the ball clears or falls short of clearing the bar, we can subtract the height of the bar from the total height reached by the ball.
Total Height = Upward Distance (h) + Initial Height of the ball
Since the ball starts at ground level, the initial height is 0. Therefore:
Total Height = Upward Distance (h)
Clearance / Fall Short = Total Height - Height of the bar
Clearance / Fall Short = 13.07 m - 3.05 m = 10.02 m
So, the ball clears the bar by approximately 10.02 meters.