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, 211) 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, 211) 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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