A place kicker must kick a football from a point 34.0 m (about 37 yd) from the goal. As a result of the kick, the ball must clear the crossbar, which is 3.05 m high. When kicked, the ball leaves the ground with a speed of 21.0 m/s at an angle of 56◦ to the horizontal.
To determine if the ball clears the crossbar, what is its height with respect to the crossbar when it reaches the plane of the crossbar? The acceleration of gravity is 9.81 m/s2 .
Answer in units of m
To solve this question, we need to find the vertical height of the ball when it reaches the horizontal plane of the crossbar. We can use the kinematic equations to solve for the height.
Step 1: Break down the initial velocity into horizontal and vertical components.
The initial velocity (v) is given as 21.0 m/s at an angle of 56 degrees to the horizontal. To find the initial vertical velocity (v_y), we can use the equation:
v_y = v * sin(θ)
where θ is the angle of inclination.
Substituting the given values:
v_y = 21.0 m/s * sin(56°)
v_y ≈ 18.102 m/s
Step 2: Determine the time of flight.
Time of flight (t) is the time taken by the ball to reach the crossbar. We can find it using the equation:
t = (2 * v_y) / g
where g is the acceleration due to gravity.
Substituting the given values:
t = (2 * 18.102 m/s) / 9.81 m/s^2
t ≈ 3.691 s
Step 3: Calculate the vertical displacement.
The vertical displacement (h) is the height of the ball with respect to the crossbar. We can calculate it using the equation:
h = v_y * t - (0.5 * g * t^2)
Substituting the given values:
h = 18.102 m/s * 3.691 s - (0.5 * 9.81 m/s^2 * (3.691 s)^2)
h ≈ 67.455 m
However, we need to consider the height above the crossbar, so we subtract the height of the crossbar (3.05 m) from the total vertical displacement:
Height with respect to the crossbar = h - 3.05 m
Height with respect to the crossbar ≈ 67.455 m - 3.05 m
Height with respect to the crossbar ≈ 64.405 m
So, the height of the ball with respect to the crossbar when it reaches the plane of the crossbar is approximately 64.405 meters.