Projectile Question: How Much Vertical Distance Does Ball Clear Crossbar?

[menesh2002]

Projectile Question?

To win the game, a place kicker must kick a
football from a point 44 m (48.1184 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 30.1◦ from the horizontal.
The acceleration of gravity is 9.8 m/s2 .
By how much vertical distance does the ball
clear the crossbar? Answer in units of m.

 

[LowlyPion]

To win the game, a place kicker must kick a
football from a point 44 m (48.1184 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 30.1◦ from the horizontal.
The acceleration of gravity is 9.8 m/s2 .
By how much vertical distance does the ball
clear the crossbar? Answer in units of m.

Welcome to PF.

What equations would you use to go about figuring it out?

 

[baba89]

I would approach this question by first analyzing the given information and then using the principles of projectile motion to determine the vertical distance the ball clears the crossbar.

To start, we know that the ball is kicked from a point 44 m away from the goal with an initial velocity of 24 m/s at an angle of 30.1◦ from the horizontal. We can use the equations of projectile motion to calculate the vertical distance the ball travels.

First, we can use the formula v = u + at to find the vertical component of the initial velocity (u) at the point of kick-off. Since the angle is given, we can use trigonometry to find the vertical component: u = 24 sin 30.1◦ = 12 m/s.

Next, we can use the formula s = ut + 1/2at^2 to find the vertical distance the ball travels in the air. Since we are only concerned with the vertical distance, we can ignore the horizontal component and focus on the vertical component. Thus, s = 12t - 1/2(9.8)t^2.

To find the time (t) it takes for the ball to reach the crossbar, we can use the formula t = (v-u)/a. Since the final velocity (v) is zero at the peak of the ball's trajectory, we can solve for t: t = u/a = 12/9.8 = 1.224 s.

Now, we can plug this value of t into the formula for s to find the vertical distance the ball travels: s = 12(1.224) - 1/2(9.8)(1.224)^2 = 7.344 m. Therefore, the ball clears the crossbar by approximately 7.344 m.

In conclusion, by using the principles of projectile motion and the given information, we can determine that the ball clears the crossbar by approximately 7.344 m. This calculation can help a place kicker to adjust their kick, if needed, to ensure that the ball clears the crossbar and leads to a successful field goal.