A place kicker must kick a football from a
point 35 m (about 38 yd) from the goal, and
the ball must clear the crossbar, which is
3.05 m high. When kicked, the ball leaves the
ground with a speed of 24 m/s at an angle of
48
◦
to the horizontal.
By how much does the ball clear or fall short
of clearing the crossbar? The acceleration of
gravity is 9.8 m/s
2
.
Answer in units of m
Doc
To find out how much the ball clears or falls short of clearing the crossbar, we need to determine the maximum height reached by the ball's trajectory.
First, we can break the initial velocity of the ball into its horizontal and vertical components. The horizontal component of the velocity remains constant throughout the motion, while the vertical component undergoes a parabolic trajectory due to the acceleration of gravity.
The initial vertical velocity (V_y0) can be calculated using the formula:
V_y0 = V * sin(θ)
where V is the initial velocity of the ball (24 m/s) and θ is the angle of the ball's trajectory (48°).
V_y0 = 24 m/s * sin(48°)
V_y0 ≈ 15.73 m/s
Next, we can calculate the time it takes for the ball to reach its maximum height. At the maximum height, the vertical velocity component becomes zero. We can use the formula:
V_y = V_y0 + a * t
where V_y is the vertical velocity, V_y0 is the initial vertical velocity, a is the acceleration due to gravity (-9.8 m/s^2), and t is the time taken.
0 = 15.73 m/s + (-9.8 m/s^2) * t
Solving for t, we get:
t = 15.73 m/s / 9.8 m/s^2 ≈ 1.60 s
Now, we can find the maximum height (H) reached by the ball using the formula:
H = V_y0 * t + (1/2) * a * t^2
H = 15.73 m/s * 1.60 s + (1/2) * (-9.8 m/s^2) * (1.60 s)^2
H ≈ 12.57 m
Therefore, the maximum height reached by the ball is approximately 12.57 meters.
To determine how much the ball clears or falls short of clearing the crossbar, we subtract the height of the crossbar (3.05 m) from the maximum height:
Clearance = H - crossbar height
Clearance = 12.57 m - 3.05 m
Clearance ≈ 9.52 m
Hence, the ball clears the crossbar by approximately 9.52 meters.