# 3. Motion in Two and Three Dimensions

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3. Motion in Two and Three Dimensions

Recap: Constant Acceleration
Area under the function v(t).

Recap: Constant Acceleration

Recap: Acceleration due to Gravity (Free Fall)
In the absence of air resistance all objects fall with the same constant acceleration of about g = 9.8 m/s2 near the Earth’s surface.

Recap: Example 2 m 5 m/s A ball is thrown upwards at 5 m/s, relative to the ground, from a height of 2 m. We need to choose a coordinate system.

Recap: Example Let’s measure v0 = 5 m/s time from when
y0 = 2 m v0 = 5 m/s y Let’s measure time from when the ball is launched. This defines t = 0. Let’s choose y = 0 to be ground level and up to be the positive y direction.

Recap: Example y0 = 2 m v0 = 5 m/s y
1. How high above the ground will the ball reach? use with a = –g and v = 0.

Recap: Example y0 = 2 m v0 = 5 m/s y 2. How long does it
take the ball to reach the ground? Use with a = –g and y = 0.

Recap: Example y0 = 2 m v0 = 5 m/s y 3. At what speed
does the ball hit the ground? Use with a = –g and y = 0.

Vectors

Vectors A vector is a mathematical quantity that has two properties:
direction and magnitude. One way to represent a vector is as an arrow: the arrow gives the direction and its length the magnitude.

Position A position p is a vector: its direction is from o
to p and its length is the distance from o to p. A vector is usually represented by a symbol like . p

Displacement A displacement is another example of a vector.

not matter, that is, vector addition is commutative.

Vector Scalar Multiplication
a and –q are scalars (numbers).

Vector Subtraction If we multiply a vector by –1 we reverse its
direction, but keep its magnitude the same. Vector subtraction is really vector addition with one vector reversed.

Vector Components Acosq is the component, or the projection, of
the vector A along the vector B.

Vector Components

Unit Vectors From the components, Ax, Ay, and Az, of a vector,
we can compute its length, A, using If the vector A is multiplied by the scalar 1/A we get a new vector of unit length in the same direction as vector A; that is, we get a unit vector.

Unit Vectors It is convenient to define unit vectors
parallel to the x, y and z axes, respectively. Then, we can write a vector A as follows:

Velocity and Acceleration Vectors

Velocity

Acceleration

Relative Motion

Relative Motion N S W E Velocity of plane relative to air
Velocity of air relative to ground Velocity of plane relative to ground

Example – Relative Motion
S W E A pilot wants to fly plane due north Airspeed: km/h Windspeed: 90 km/h direction: W to E 1. Flight heading? 2. Groundspeed? Coordinate system: î points from west to east and ĵ points from south to north.

Example – Relative Motion
S W E

Example – Relative Motion
S W E

Example – Relative Motion
S W E Equate x components 0 = –200 sinq + 90 q = sin-1(90/200) = 26.7o west of north.

Example – Relative Motion
S W E Equate y components v = 200 cosq = 179 km/h

Projectile Motion

Projectile Motion under Constant Acceleration
Coordinate system: î points to the right, ĵ points upwards

Projectile Motion under Constant Acceleration
Impact point R = Range

Projectile Motion under Constant Acceleration
Strategy: split motion into x and y components. R = Range R = x - x0 h = y - y0

Projectile Motion under Constant Acceleration
Find time of flight by solving y equation: And find range from:

Projectile Motion under Constant Acceleration
Special case: y = y0, i.e., h = 0 R y0 y(t)

Uniform Circular Motion

Uniform Circular Motion

Uniform Circular Motion
Velocity

Uniform Circular Motion
dq/dt is called the angular velocity

Uniform Circular Motion
Acceleration For uniform motion dq/dt is constant

Uniform Circular Motion
Acceleration is towards center Centripetal Acceleration

Uniform Circular Motion
Magnitudes of velocity and centripetal acceleration are related as follows

Uniform Circular Motion
Magnitude of velocity and period T related as follows r

Summary In general, acceleration changes both the magnitude and direction of the velocity. Projectile motion results from the acceleration due to gravity. In uniform circular motion, the acceleration is centripetal and has constant magnitude v2/r.

How to Shoot a Monkey H x = 50 m h = 10 m H = 12 m Compute minimum
initial velocity H