Okay, here's the problem:
During a football game, the quarterback rolls out to his right with two huge linebackers in close pursuit. An instant before the linebackers bury him at the 50 yard line (46-m), he releases a towering pass down the sideling. The ball sails toward the end zone with a horizontal velocity of 11.4m/s. At the instant the pass is released by the quarterback, the offensive receiver is crossing the 40-yard (36m) line, along the sideline at a velocity of 8.8 m/s. The defensive safety is 2-m behind the offensive receiver and moving down the sideline at a velocity of 9.5 m/s. The pass reaches a maximum height of 21.6m and is caught in the end zone. Determine if the football is caught for a touchdown or an interception as follows. (Assume no air resistance.)
Now, I drew a picture so that I could better understand what was going on. But I don't quite get what "The pass reaches a maximum height of 21.6m and is caught in the end zone." means. I don't understand how I would use that in the problem. Can anyone explain that please?
I'm going to post the question in parts to make it less confusing. Thanks in advance for the help!
This is the height of silliness, I would drop the class. All this violent reality, then the discamimer neglect air resistance. Assume Santa and the North pole exists, but forget about the reindeer. Whatever. We ought not be burying folks on a playing field.
The issue seems to be is whom get to the ball first.
So, figure the time the ball is in the air, use the max height to do this.
Then, knowing time in air, find where it lands at thown height (which is catch height). (We call that horizontal distance given an initial velocity and time).
Then using your sketch, note the receiver travels that distance less ten meters. Does he travel that distance (time= distance/hisSpeed) in the same time the ball is in the air?
If not, the defensive guy travels the distance the ball is in the air (46m-36m+2m) (time=distance/hisspeed) and he catches it.
There is some question in my mind where the defensive guy is: "is 2 m behind the receiver" could mean differing things. Normally that would mean the defensive guy is 2 m closer to the endpost, but I don't think that is the intent here, but I don't know.
To determine whether the football is caught for a touchdown or an interception, you need to analyze the motion of the ball, the offensive receiver, and the defensive safety.
First, let's address your question about the maximum height of the pass. The maximum height of 21.6m is the highest point the football reaches during its trajectory. This information is important because it allows us to determine the time it takes for the ball to reach its highest point, which in turn helps us calculate the total time the ball is in the air.
Now, let's break down the problem into steps:
1. Calculate the time the ball is in the air:
Since there is no air resistance, the motion of the ball is symmetrical. The time it takes for the ball to reach its maximum height is equal to the time it takes to descend from the maximum height to the ground. You can use the following kinematic equation to find this time:
H = (1/2) * g * t^2
where H is the maximum height (21.6m), g is the acceleration due to gravity (9.8 m/s^2), and t is the time in seconds.
Solve this equation for t to find the duration the ball remains in the air.
2. Calculate the horizontal distance the ball travels:
Use the horizontal velocity of the ball (11.4 m/s) and the time calculated in step 1 to find the distance the ball covers horizontally. Since the motion is straight and horizontal, the horizontal distance is given by:
Distance = Velocity * Time
Compute this distance to determine where the ball lands (catch height).
3. Analyze the receiver's position:
From the given information, we know the receiver is crossing the 40-yard line (36m) at a velocity of 8.8 m/s. To determine whether the receiver reaches the catch height (where the ball lands) in the same time the ball is in the air, calculate the time it takes for the receiver to cover the horizontal distance between their starting position and the landing position of the ball. Use the equation:
Time = Distance / Velocity (where Distance is the difference between the receiver's starting position and the landing position)
4. Analyze the defensive safety's position:
The defensive safety is initially 2m behind the receiver and moving down the sideline at a velocity of 9.5 m/s. Determine if the defensive safety reaches the catch height (where the ball lands) in the same time the ball is in the air. Calculate the time it takes for the defensive safety to cover the horizontal distance between their starting position and the landing position of the ball using the equation:
Time = Distance / Velocity (where Distance is the sum of the initial distance between the defensive safety and the receiver and the additional distance the defensive safety needs to cover to reach the landing position)
5. Compare the times:
Compare the time it takes for the receiver to reach the catch height with the time it takes for the defensive safety to reach the catch height. If the receiver's time is equal to or less than the defensive safety's time, then the football is caught for a touchdown. If the defensive safety's time is less, then it is an interception.
Remember to consider the interpretation of "2m behind the receiver" in step 4. If it means the defensive safety is 2m closer to the end zone than the receiver, adjust the calculations accordingly.
By going through these steps, you can determine whether the football is caught for a touchdown or an interception based on the given information.