Kinematics in Two Dimensions Notes

Kinematics in Two Dimensions

Projectile Motion

Projectile motion is the motion of an object fired at an angle \theta with the horizontal.
This motion can be analyzed by considering the horizontal (x) and vertical (y) components of the motion independently.
If air resistance is negligible, the horizontal component of motion does not change, thus the horizontal component has constant velocity.
The horizontal motion of a projectile is described by the equation: x = x0 + vx t where vx = v0 \cos \theta.
The initial vertical velocity is given by v{0y} = v0 \sin \theta.
The vertical component of motion is affected by gravity, resulting in uniform constant acceleration motion.
The vertical motion is described by the equations for an object in free fall:
- vy = v{0y} - gt
- y = y0 + v{0y} t - \frac{1}{2} g t^2
- vy^2 = v{0y}^2 - 2g(y - y_0)
- (vy + v{0y})t = 2(y - y_0)

Problem 1

A spring gun on a table fires a steel ball horizontally. The ball starts 1.0 m above the floor and travels 2.7 m horizontally before hitting the floor. Find:

a) the time the ball is in the air
b) the initial velocity of the ball

Problem 2

A spring gun on a table fires a steel ball at a 45^\circ angle above the horizontal. The ball leaves the muzzle 1.1 m above the floor and travels 4.6 m horizontally. Determine:

a) the total time that the ball is in the air.

Problem 3

A projectile is fired with an initial speed of 113 m/s at an angle of 60.0^\circ above the horizontal from the top of a cliff 49.0 m high. Determine:

a) the time to reach the maximum height
b) the maximum height above the base of the cliff reached by the projectile

Problem 4

A stone is thrown horizontally outward from the top of a bridge 19.6 m above the street below. The initial velocity of the stone is 5.0 m/s. Determine:

a) the total time that the stone is in the air
b) the magnitude and direction of the velocity of the projectile just before it strikes the street

Relative Velocity

Relative velocity is the velocity of an object with respect to a particular frame of reference.
The velocity of an object relative to one frame of reference can be found by vector addition if its velocity relative to a second frame of reference and the relative velocity of the two reference frames are known.
The equation for relative velocity is given by: \vec{V}{BS} = \vec{V}{BW} + \vec{V}_{WS}
- \vec{V}_{BS} is the velocity of the Body with respect to the Shore.
- \vec{V}_{BW} is the velocity of the Body with respect to the Water.
- \vec{V}_{WS} is the velocity of the Water with respect to the Shore.