Consider an object moving in two dimensions whose position vector at time t is given by...

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Consider an object moving in two dimensions whose position vector at time t is given by 7 (t) = (-1², 21-1) for t 20 (a) Find expressions for the velocity vector v(t), ds the acceleration vector ä(t), and the speed dt +(1) (b) Find an expression for the tangential component of acceleration a, by applying the d's formula a, = dt (c) Find an expression for the unit tangent vector f(t). (ctd) (d) Find an expression for the tangential component of acceleration a, by applying the formula a, =ā Ť. Your answer should agree with part (b). (e) Compute ā (1), f(1), and a, (1). á (1) = f(1) = (f) Find a,, (1) using the formula a, (1)=√(1)-[a (1)]² a, (1) = g) Find (1). (This should not require any tedious computation; note that in 2 dimensions, the vectors (-b, a) and (b.-a) are the only two vectors that are both orthogonal to (a, b) and equal in length to (a, b). This makes it easy to find N if † is already known, provided you can figure out which of these two vectors to choose. Look at the figure at the beginning of the problem for guidance; remember that N always points toward the concave side of the curve.) (h) Using the formula a(t)= a,f(t)+a,Ñ(t), use the answers you have obtained thus far to express a (1) as a linear combination of f(1) and N(1). As a final check on your work, the two sides of the resulting equation should be in agreement. Consider an object moving in two dimensions whose position vector at time t is given by 7 (t) = (-1², 21-1) for t 20 (a) Find expressions for the velocity vector v(t), ds the acceleration vector ä(t), and the speed dt +(1) (b) Find an expression for the tangential component of acceleration a, by applying the d's formula a, = dt (c) Find an expression for the unit tangent vector f(t). (ctd) (d) Find an expression for the tangential component of acceleration a, by applying the formula a, =ā Ť. Your answer should agree with part (b). (e) Compute ā (1), f(1), and a, (1). á (1) = f(1) = (f) Find a,, (1) using the formula a, (1)=√(1)-[a (1)]² a, (1) = g) Find (1). (This should not require any tedious computation; note that in 2 dimensions, the vectors (-b, a) and (b.-a) are the only two vectors that are both orthogonal to (a, b) and equal in length to (a, b). This makes it easy to find N if † is already known, provided you can figure out which of these two vectors to choose. Look at the figure at the beginning of the problem for guidance; remember that N always points toward the concave side of the curve.) (h) Using the formula a(t)= a,f(t)+a,Ñ(t), use the answers you have obtained thus far to express a (1) as a linear combination of f(1) and N(1). As a final check on your work, the two sides of the resulting equation should be in agreement.

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Lets break this down step by step a Find expressions for the velocity vector overlinevt the accelera... View the full answer