A football is kicked off with an initial speed of 65 ft/s at a projection angle of 45 degrees. A receiver on the goal line 63 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?
To calculate the minimum speed required for the receiver to catch the ball before it hits the ground, we need to consider the horizontal and vertical components of the ball's motion.
Let's start by breaking down the initial velocity of the ball into its horizontal and vertical components:
Horizontal Component: Vx = V * cos(θ)
Vertical Component: Vy = V * sin(θ)
where V is the initial speed of the ball (65 ft/s) and θ is the projection angle (45 degrees).
Horizontal Component: Vx = 65 ft/s * cos(45)
Vertical Component: Vy = 65 ft/s * sin(45)
Vx = 65 ft/s * (√2/2)
Vy = 65 ft/s * (√2/2)
Vx = 45.96 ft/s
Vy = 45.96 ft/s
Since we are interested in the minimum speed required for the receiver to catch the ball before it hits the ground, we only need to consider the vertical component of the ball's motion.
Now, let's determine the time it takes for the ball to reach the receiver's position, which is 63 yards away from the point of projection.
First, convert 63 yards to feet:
63 yards * 3 feet/yard = 189 feet
Next, we can use the equation of motion to calculate the time (t) it takes for the ball to reach the receiver's position:
Vy = gt
where g is the acceleration due to gravity (approximately 32 ft/s^2).
45.96 ft/s = 32 ft/s^2 * t
Solving for t:
t = 45.96 ft/s / 32 ft/s^2
t ≈ 1.4365 seconds
Now that we have the time it takes for the ball to reach the receiver's position, we can determine the minimum speed required for the receiver to catch the ball before it hits the ground.
The minimum speed required can be calculated using the following equation:
Minimum Speed = Distance / Time
where Distance is the distance between the receiver and the point of projection (189 feet), and Time is the previously calculated time (1.4365 seconds).
Minimum Speed = 189 ft / 1.4365 s
Minimum Speed ≈ 131.48 ft/s
Therefore, the receiver must run at a minimum speed of approximately 131.48 ft/s in order to catch the ball before it hits the ground.