Consider a linear robot of length of 10 units, that is moving along the line x = 10 at a constant velocity of 1 m/s. It starts from (10, 0) at t = 0, and is dribbling a ball forward (into the 1st quadrant) along this straight path. There is no obstruction or robot present in this half of the field. It needs to shoot the ball in a goal, having two goalposts at (-2, 30) and (2, 30).Draw a figure for the situation
Is there a specific time at which the robot can shoot the ball (along the ground in a straight line) to have the maximum probability of putting the ball in the goal ? (Hint: Note here that the robot can shoot at any point inside the two goalposts)
How does your answer for the optimal time change if the starting position of the robot changes from (10, 0) to (x, 0) and everything else remains the same ?
How does your answer for the optimal time change if the width of the goal increases, i.e. changes from (-2 30), (2, 30) to (-a, 30), (a, 30), and everything else remains the same ?
Consider that there is now a ‘defending’ robot in the form of a line. The ‘line’ of this defending robot extends from (-2, 15) to (2, 15). (bang in front of the (-2, 30) to (2, 30) goal). Now our attacking robot decides to shoot from (10, 0) instead of moving forward and risk losing the ball to the defender. And intelligently, it decides to shoot the ball at the wall such that it reflects into the goal (this is allowed). What is the range of y-values along the two walls that the robot should shoot the ball in to score?
Bonus*: Can you create a simulation for the situation in part a with one modification - the user should be able to define a start coordinate for a robot along x - axis (instead of it being fixed at (10, 0)). Then an animation should run that shows the robot moving straight ahead from that point, shooting the ball at the optimal time, stopping and then printing the time.
Answers
Explanation:
Consider that there is now a ‘defending’ robot in the form of a line. The ‘line’ of this defending robot extends from (-2, 15) to (2, 15). (bang in front of the (-2, 30) to (2, 30) goal). Now our attacking robot decides to shoot from (10, 0) instead of moving forward and risk losing the ball to the defender. And intelligently, it decides to shoot the ball at the wall such that it reflects into the goal (this is allowed). What is the range of y-values along the two walls that the robot should shoot the ball in to score?
Bonus*: Can you create a simulation for the situation in part a with one modification - the user should be able to define a start coordinate for a robot along x - axis (instead of it being fixed at (10, 0)). Then an animation should run that shows the robot moving straight ahead from that point, shooting the ball at the optimal time, stopping and then printing the time
Answer:
15.0kg[25.0km(vy)^k]=15.0kg[2.50×104m(−2.0×103m/s−(2.0m/s2)t)^k].
Bonus*: Can you create a simulation for the situation in part a with one modification - the user should be able to define a start coordinate for a robot along x - axis (instead of it being fixed at (10, 0)). Then an animation should run that shows the robot moving straight ahead from that point, shooting the ball at the optimal time, stopping and then printing the time