A place kicker must kick a football from a point 32.2 m from a 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 20.8 m/s at an angle of 53° 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?
falling
rising
To determine how much the ball clears or falls short of clearing the crossbar and whether it approaches the crossbar while still rising or falling, we can use the principles of projectile motion.
Projectile motion refers to the motion of an object launched into the air at an angle. In this case, the football is kicked from the ground at an angle of 53° to the horizontal with an initial speed of 20.8 m/s.
First, we need to break down the initial velocity components of the ball:
- Vertical component: V_y = V * sin(θ)
- Horizontal component: V_x = V * cos(θ)
Given:
- Initial speed, V = 20.8 m/s
- Angle, θ = 53°
- Distance to the goal, 32.2 m
- Height of the crossbar, 3.05 m
(a) To calculate how much the ball clears or falls short of clearing the crossbar, we need to determine the vertical motion of the ball when it reaches the goal.
We can find the time it takes for the ball to reach the goal using the horizontal component of the velocity:
Time, t = Distance / Horizontal component velocity
= 32.2 m / (20.8 m/s * cos(53°))
Next, we need to calculate the vertical position of the ball when it reaches the goal using the vertical component of the velocity. This will tell us how much the ball clears or falls short of clearing the crossbar:
Vertical position = Vertical component velocity * time - (0.5 * g * t^2)
= V_y * t - (0.5 * 9.8 m/s^2 * t^2)
Substitute the values and calculate to find the vertical position.
Finally, compare the height of the crossbar with the calculated vertical position to determine if the ball clears or falls short of the crossbar.
(b) To determine whether the ball approaches the crossbar while still rising or while falling, we can analyze the vertical motion of the ball at the goal.
If the vertical component velocity is greater than 0 when the ball reaches the goal, it is still rising. If the vertical component velocity is less than 0, then it is falling.
By calculating the vertical component velocity at the goal using the formula:
Vertical component velocity = V_y - g * t
Substitute the values and calculate the vertical component velocity.
If the result is positive, then the ball approaches the crossbar while still rising. If the result is negative, then the ball approaches the crossbar while falling.