Physics Exercises/Kinematics in 2-3 Dimensions

Kinematics in 2 or 3 dimensions is essentially the same as ../Kinematics in One Dimension/ with an extra step added: one must separate the motion into 2 or 3 perpendicular directions (for each problem, one particular choice of axes is better than others) and solve motion in each direction separately as it is done in ../Kinematics in One Dimension/.

Without Calculus

 * 1) A cannon (assumed to be at ground level) is fired off a flat surface at an angle θ above the horizontal with an initial speed of $$v_0$$. Assume no air resistance.
 * 2) What is the initial horizontal component of the velocity?
 * 3) What is the initial vertical component of the velocity?
 * 4) What is the horizontal component of the velocity at the highest point of the trajectory?
 * 5) What is the vertical component of the velocity at that point?
 * 6) What is the horizontal component of the velocity when the projectile lands?
 * 7) What is the vertical component of the velocity when it lands?
 * 8) A woman walks north at 3 km/h on a boat that is moving east at 4 km/h.
 * 9) With respect to the shore, how fast is the woman moving?
 * 10) With respect to the shore, at what angle from the north is she moving?

With Calculus

 * 1) Just for this problem, assume t is a unitless quantity measuring time. A cockroach has a velocity $$v = \sin(t)~\hat{x} + e^t~\hat{y}$$. The cockroach starts at the origin. Derive an expression for
 * 2) the speed of the cockroach as a function of time,
 * 3) the position of the cockroach as a function of time,
 * 4) the acceleration of the cockroach as a function of time.