question archive Denote by q (t) is the position at time t of an object moving on a line
Denote by q (t) is the position at time t of an object moving on a line
Subject:MathPrice: Bought3
Denote by q (t) is the position at time t of an object moving on a line. The function q need not be a position, q may be any differentiable function. The derivative, q, of q need not be a velocity, q is just the derivative of q. Assume that q is differentiable, so that its derivative, q = q', is the velocity, v: v (t) = q(t) = q' (t) (1) Position and velocity are given, or measured, at three times, to, t1, and t2, with to < t1 < t2 (2) v (to) = vo > v1 = v (t1) > v2 = v (t2) 2 0 (3) 0 < q(to) = 90 < q1 = q (t1) < 92 = q(t2) (4) Assume also that the position increases as the velocity decreases on the interval of time [to, t2]. The problem is to establish a procedure to estimate a time t, between to and t1 when the position q. = q (t.) is exactly halfway between go and q1: q. = q(t.) = 40 + 91 (5) 2 Assume that v is affine: for some constant a, v (t) = vo + (t -to) . a (6) (1.1) Calculate a in terms of to, t1, vo and v1- (1.2) Find a formula for q (t) for every t in terms of t, to, t1, 90, 91, Vo, V1- (1.3) Then solve Equation (5) for t. in terms of to, t1, 90, 91, Vo, and v1- (1.4) Calculate the velocity when the position is halfway, v (t.)
