Motion of a mass in response to applied force. A unit mass moves on a straight line (in one dimension). The position of the mass at time t (in seconds) is denoted by s(t), and its derivatives (the velocity and acceleration) by s ±(t) and s ±±(t). The position as a function of time can be determined from Newton’s second law
where F(t) is the force applied at time t, and the initial conditions s(0), s ± (0). We assume F(t) is piecewise-constant, and is kept constant in intervals of one second. The sequence of forces F(t), for 0 ≤ t <>
Derive expressions for the final velocity s ± (10) and final position s(10). Show that s(10) and s ± (10) are affine functions of x, and give 10-vectors a, c and constants b, d for which
 This means that the mapping from the applied force sequence to the final position and velocity is affine. Hint. You can use
You will find that the mass velocity s ‘(t) is piecewise-linear.
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