The top of the pool table is 3.05 ft from the floor. The placement of the tape is such that 0 feet aligns to the edge of the table (as shown). The winner of the competition wants to know if he has broken the world record for the break shot of 32 mph. His ball landed a distance 12.25 ft from the table edge. Calculate his break shot speed in mph and at what speed did his ball hit the ground (in mph)
find the flight time
... 0 = -16 t^2 + 3.05
break shot speed
... 12.25 ft / flight time
(impact speed)^2 =
... (break shot speed)^2 + (t * 32 ft/s^2)^2
To calculate the break shot speed in mph, we can use the formula:
Speed (mph) = Distance / Time
First, let's determine the time it took for the ball to travel the distance of 12.25 ft from the table edge. To do this, we need to find the time it takes for an object to fall from a height of 3.05 ft.
The time it takes for an object to fall can be calculated using the equation:
Time = √(2h / g)
Where:
h is the height
g is the acceleration due to gravity (approximately 32.174 ft/s²)
Plugging in the values:
h = 3.05 ft
g = 32.174 ft/s²
Time = √(2 * 3.05 ft / 32.174 ft/s²)
Time = √(0.1894 s²)
Time = 0.4353 s (approximately)
Now that we have the time, we can calculate the break shot speed in mph:
Break Shot Speed = Distance / Time
Break Shot Speed = 12.25 ft / 0.4353 s
To convert the speed from feet per second (fps) to miles per hour (mph), we need to multiply it by the conversion factor:
1 mph = 1.467 ft/s
So, Break Shot Speed (mph) = (12.25 ft / 0.4353 s) * 1.467 ft/s
Break Shot Speed (mph) = 32.68 mph (approximately)
Therefore, the break shot speed of the winner is approximately 32.68 mph.
To calculate the speed at which his ball hit the ground, we assume the ball was launched horizontally. In this case, the vertical component of the ball's initial velocity is due to its fall from the table height.
Using the equation of motion:
Vertical Distance = (1/2) * g * t²
Where:
g is the acceleration due to gravity (32.174 ft/s²)
t is the time (0.4353 s)
Vertical Distance = (1/2) * 32.174 ft/s² * (0.4353 s)²
Vertical Distance = 3.299 ft (approximately)
Since the vertical distance of the ball hitting the ground is 3.299 ft, we can calculate its speed using:
Speed = √(2 * g * Vertical Distance)
Speed = √(2 * 32.174 ft/s² * 3.299 ft)
Speed = √(213.698 ft²/s²)
Speed = 14.625 ft/s (approximately)
To convert from feet per second to miles per hour, multiply the speed by the conversion factor:
1 mph = 1.467 ft/s
Ball Speed at Ground (mph) = 14.625 ft/s * 1.467 ft/s
Ball Speed at Ground (mph) = 21.45 mph (approximately)
Therefore, the ball hit the ground at approximately 21.45 mph.