Homework H1.E - Sp23

Problem statement
Solution video



As you watch particle P move through its path, focus on a few things:

  • Direction of the velocity vector. The velocity of the ball is always tangent to the path, as expected. Note that the unit tangent vector e_t points in the same direction as the velocity.
  • Direction of the acceleration vector. The acceleration of the P always points "inward" on the path, again, as expected. Here you see that the acceleration vector points in the positive e_t direction; therefore, its speed must be increasing with time (as it is).
  • Directions of the polar unit vectors. The e_r unit vector always points outward from  point O toward P. The e_θ unit vector is perpendicular to e_r, and in the direction of increasing θ, as seen in the animation above.

Your task is to calculate R and θ, and their time derivatives. Here, you are given R and θ explicitly as a function of time. Therefore, the chain rule of differentiation is not needed.

Any questions?? Please ask/answer questions regarding this homework problem through the "Leave a Comment" link above.

24 thoughts on “Homework H1.E - Sp23”

    1. Since the problem does not specify which set of coordinates that are expected for the final answer, you are free to choose either one. My recommendation would be to keep it in terms of the polar unit vectors, as that would be less work for you.

    1. I'm pretty sure they are asking us to find the vectors of velocity and acceleration. They would tell us if they wanted us just to find the scalar of each.

  1. Were are given a value for t. Does this mean we should give our answers for velocity and acceleration at time=4s or should we keep them as a function of t?

    1. Yes, they need to be the same. Each component of velocity must have units of m/s. Each component of acceleration must have units of m/s^2. Otherwise, you cannot add together the components (it is an apples-and-oranges thing).

  2. To do the derivatives of the formula 'r', can we just use the formula given in the question, or do we need to plug it into the formula r-hat = r*er-hat?

    1. Im pretty sure you first take the derivative, then use the two functions discussed in class to turn it into the forms with er-hat and etheta-hat.

  3. Since we are given r and theta in terms of t, can we just take the r_dot and theta_dot to find the components of the velocity vector and then r_dot_dot and theta_dot_dot for the acceleration vectors or am I missing something?

      1. My assumption is that we would use the equations. Take the derivative of the given r and theta equations twice, plug in t=4sec and substitute into the equations.

  4. Did anyone else find that acceleration didn't have a tangential component and velocity didn't have an er vector? Since both these unit vectors must be perpendicular it would make sense but, I am concerned that this may be due to improper differentiation.

    1. If you plugged in the equation for theta and then solved for vbar = rdot * er + r * thetadot * etheta . you would get both er and e theta components for velocity. similarly solving the equation for acceleration would yield both er and etheta components

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