Question

You have been tasked with building a robot that plays table tennis (i.e., ping pong). A ping pong table is 280 cm in length and is 150 cm wide. There is a net stretched across the center of the table, 140cm in from each side. The robot can move side to side with a velocity of 1000 cm/s. The robot works by capturing 3 pictures of the ball in motion; these 3 pictures all should be taken before the ball breaks the plane of the net, which is halfway across the table. It then will process the images and determine the optimal location that it must be to hit the ball. It is possible that that optimal location may be all the way to the left or all the way to the right of the 150 cm side. Given that all 3 pictures must be captured, and the physics of the movement of the robot (you may assume it can instantaneously change from 0 to 1000 cm/s of velocity, and a maximum ping pong ball speed of 20 miles per hour, how long can the calculations be between the 3 pictures being taken and the robot commencing motion if it is guaranteed to hit the ball? In solving this problem, sketch your solution and show your calculations.

Answer 1

To determine the maximum time allowed for calculations between the three pictures being taken and the robot commencing motion, we need to consider the time it takes for the ball to cross the net and reach the robot's optimal location.

Let's analyze the scenario step by step:

1. Ball crossing the net:
The net is located 140 cm from each side, so the distance the ball needs to travel before crossing the net is \(280 \, \text{cm} - 140 \, \text{cm} = 140 \, \text{cm}\).
The maximum speed of the ball is given as 20 miles per hour. Let's convert this speed to centimeters per second:
\(20 \, \text{miles/hour} = 20 \times 1609.34 \, \text{cm/hour} \approx 32,187 \, \text{cm/hour} \approx 8.94 \, \text{cm/second}\).
Therefore, the time it takes for the ball to cross the net is \(140 \, \text{cm} / 8.94 \, \text{cm/second} \approx 15.67 \, \text{seconds}\).

2. Robot's movement:
The robot can move side to side with a velocity of 1000 cm/s. To reach the optimal position, the robot needs to cover a distance of either 150 cm or 130 cm, depending on its initial position.
The maximum time it takes for the robot to move to either side is \(150 \, \text{cm} / 1000 \, \text{cm/second} = 0.15 \, \text{seconds}\).

3. Calculations time:
The maximum time allowed for calculations is the time remaining after subtracting the time for the ball to cross the net and the time for the robot's movement from the total time available.
Total time available: 15.67 seconds
Time for ball crossing the net: 15.67 seconds
Time for robot's movement: 0.15 seconds
Calculations time = Total time available - Time for ball crossing the net - Time for robot's movement.

Calculations time = 15.67 seconds - 15.67 seconds - 0.15 seconds = 0.85 seconds.

Therefore, the maximum time allowed for calculations between the three pictures being taken and the robot commencing motion is approximately 0.85 seconds.