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Motion in Two Dimensions

+Projectile Motion

Two-dimensional motion is called projectile motion.

Objects that are thrown or launched into the air and are subject to gravity are called projectiles.

Some examples of projectiles are

Softballs,

Footballs

Arrows

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The parabolic path which is common for all projectile motion is called trajectory.

The horizontal distance covered by a projectile which returns its original height is called the range of the projectile.

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An object is projected with an initial velocity, vi, at an angle of θ. Resolve the initial velocity into its x and y components. Then, the kinematic equations can be applied to describe the motion of the projectile throughout its flight.

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Suppose the initial velocity vector makes an angle θ with the horizontal. Again, to analyze the motion of such a projectile, you must resolve the initial velocity vector into its components.

vx,i = vi cosθ and vy,i = vi sinθ

+Example:

(a) Without air resistance, the soccer ball would travel along a parabola. (b) With air resistance, the soccer ball would travel along a shorter path.

+Example:

(a) A long jumper’s velocity while sprinting along the runway can be represented by a horizontal vector.(b) Once the jumper is airborne, the jumper’s velocity at any instantcan be described by the components of the velocity.

+ We can substitute these values for v0x and v0y

into the kinematic equations to obtain a set of equations that can be used to analyze the motion of a projectile launched at an angle.

v0x=v0cosθx = v0xt = v0cosθt v0y=v0sinθvy = v0y – gt Δy =v0yt – (1/2)gt2

Δy = (1/2)(v0y+vy)tvy

2 = v0y2 – 2gΔy

v2 = vx2 + vy

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For the motion on x axis

For the motion on y axis

The relation between speeds

+Example

A ball is launched at a velocity of 8 m/s and an angle of 53o with the horizontal line axis.

a) Calculate the time for the ball to reach its max. height.

b) How high will the ball rise?

c) What is the range of the ball?

d) What is the ball’s velocity just before it strikes the ground?

+Example

A rocket is launched at a velocity of 25 m/s and an angle of 37o with the horizontal line axis.

a) Calculate the time for the ball to reach its max. height.

b) How high will the ball rise?

c) What is the range of the ball?

d) What is the ball’s velocity just before it strikes the ground?