A rubber ball of mass \(M\) is moving with speed \(v\) toward a vertical wall. The ball's velocity vector makes an angle of \(45^\circ\) with the wall's surface. The ball strikes the wall and rebounds such that its new speed is \(\dfrac{v}{2}\) at an angle of \(45^\circ\) relative to the wall's surface. A coordinate system is defined such that the positive \(x\)-direction is directed away from the wall and the positive \(y\)-direction is parallel to the wall in the direction of the ball's initial motion. What is the impulse vector \(\vec{J}\) exerted by the wall on the ball?

Consider the following definitions of the unit vectors: \(\hat{i}\) is in the positive \(x\)-direction and \(\hat{j}\) is in the positive \(y\)-direction.