Skip to main content
Back

Instantaneous Velocity in 2D quiz

Control buttons has been changed to "navigation" mode.
1/15
  • How do you calculate instantaneous velocity from a position function r(t) in 2D?
    Take the derivative of the position function with respect to time: v(t) = dr/dt.
  • What mathematical operation do you use to find the position function from a velocity function in 2D?
    You use the indefinite integral of the velocity function and add an integration constant.
  • When integrating a velocity function to find position, what must you remember to include?
    You must include the integration constant, which is determined by initial conditions.
  • How do you calculate displacement in 2D using velocity?
    Calculate the definite integral of the velocity function from the initial to final time, or subtract the initial position from the final position.
  • What is the difference between the indefinite and definite integral in this context?
    The indefinite integral gives the position function with an integration constant, while the definite integral gives displacement without the constant.
  • How do you determine the integration constant when finding position from velocity?
    Plug the given initial conditions into your position function and solve for the constant.
  • What is the formula for displacement Δr in terms of position vectors?
    Δr = r(t_final) - r(t_initial).
  • How do you take the derivative of a vector function with i and j components?
    Take the derivative of each component separately and keep the i and j unit vectors.
  • What happens to constant terms when you take the derivative with respect to time?
    Constant terms become zero when differentiated with respect to time.
  • How do you integrate a constant vector component, such as 5j, with respect to time?
    Integrate as you would a constant: it becomes 5t in the j direction.
  • If v(t) = 3t^2 i + 5j, what is the indefinite integral with respect to t?
    The integral is t^3 i + 5t j plus the integration constant C.
  • How do you evaluate a definite integral for displacement between two times?
    Find the antiderivative, evaluate it at the final and initial times, and subtract: F(t_final) - F(t_initial).
  • Why do you not include the integration constant when calculating displacement with a definite integral?
    Because the constant cancels out when subtracting the initial value from the final value.
  • What is the result of integrating v(t) = 3t^2 i + 5j from t = 2 to t = 4?
    The displacement vector is 56i + 10j.
  • Why do the procedures for derivatives and integrals in 2D vectors resemble those in 1D?
    Because the operations are performed component-wise, just as in 1D, but with i and j unit vectors.