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: $x = x<em>0 + v</em>x t$ where $v<em>x = v</em>0 \cos \theta$.
• The initial vertical velocity is given by $v<em>{0y} = v</em>0 \sin \theta$.
• 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:
  • $v<em>y = v</em>{0y} - gt$
  • $y = y<em>0 + v</em>{0y} t - \frac{1}{2} g t^2$
  • $v<em>y^2 = v</em>{0y}^2 - 2g(y - y_0)$
  • $(v<em>y + v</em>{0y})t = 2(y - y_0)$

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: $\vec{V}<em>{BS} = \vec{V}</em>{BW} + \vec{V}_{WS}$
  • $\vec{V}_{BS}$ 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.

Problem 5

A woman swims perpendicular across a river at 2.0 m/s with respect to the water. The river is 300 m wide and the current is 1.0 m/s. Determine:

• a) the woman's velocity relative to the shore,
• b) the distance swept downstream, and
• c) the time required to swim across the river.