A football is kicked off with an initial speed of 63 ft/s at a projection angle of 45 degrees. A receiver on the goal line 67 yd away in the direction of the kick starts running to meet the ball at that instant. What must be his minimum speed (in feet/second) if he is to catch the ball before it hits the ground?

Question

A football is kicked off with an initial speed of 63 ft/s at a projection angle of 45 degrees. A receiver on the goal line 67 yd away in the direction of the kick starts running to meet the ball at that instant. What must be his minimum speed (in feet/second) if he is to catch the ball before it hits the ground?

I've tried many different things and cannot figure this out. If anyone has time can you please walk me through this problem? It will help me on my others I have to do as well.

Answer 1

See the steps I gave you. Start with those, I will check your work on each step.

Answer 2

To solve this problem, we need to consider the horizontal and vertical components of the football's motion separately.

First, let's determine the time it takes for the football to reach the receiver on the goal line. Since the football is kicked off at an angle of 45 degrees with an initial speed of 63 ft/s, we can use the following formula:

Time of flight = (2 * initial speed * sin(angle of projection)) / acceleration due to gravity

Plugging in the given values:

Time of flight = (2 * 63 ft/s * sin(45 degrees)) / (32.2 ft/s^2)

Calculating this, we find that the time of flight is approximately 2.03 seconds.

Now, let's find the horizontal distance covered by the football during this time. The horizontal component of the initial velocity will remain constant throughout the motion, so we can use the equation:

Horizontal distance = initial speed * cos(angle of projection) * time

Again, plugging in the given values:

Horizontal distance = 63 ft/s * cos(45 degrees) * 2.03 s

Calculating this, we find that the horizontal distance covered by the football is approximately 86.84 ft.

Since the receiver is 67 yd (201 ft) away, we can now determine the minimum speed the receiver must have to catch the ball before it hits the ground. The receiver needs to cover this distance during the same time it takes for the ball to reach the goal line (2.03 seconds). Therefore, we can use the equation:

Minimum speed = horizontal distance / time of flight

Substituting the values:

Minimum speed = 201 ft / 2.03 s

Calculating this, we find that the receiver must have a minimum speed of approximately 99 ft/s in order to catch the ball before it hits the ground.

I hope this step-by-step explanation helps you understand how to solve this problem.