# 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.
1. What is the initial horizontal component of the velocity?
2. What is the initial vertical component of the velocity?
3. What is the horizontal component of the velocity at the highest point of the trajectory?
4. What is the vertical component of the velocity at that point?
5. What is the horizontal component of the velocity when the projectile lands?
6. What is the vertical component of the velocity when it lands?
2. A woman walks north at 3 km/h on a boat that is moving east at 4 km/h.
1. With respect to the shore, how fast is the woman moving?
2. 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
1. the speed of the cockroach as a fuction of time,
2. the position of the cockroach as a function of time,
3. the acceleration of the cockroach as a function of time.