Topic 1.3 Extended B - Components of motion

Up to now we have considered objects moving in one dimension. However, most objects move in more than one dimension.For example, consider the ball shown here:

Motion in Two Dimensions3-1 Components of motion

We can sketch in our x and y for successive snapshots to obtain an idea of the different velocities the ball has at different times: x is in YELLOW. y is in RED. We can also sketch in the displacement d of the ball at each time interval (in GREEN).Let's examine one time interval in detail:

x

y

d

y

FYI: The displacement vector gives the direction of the motion

From the Pythagorean Theorem we can find the value of d if we know x and y:d2 = x2 + y2

Topic 1.3 Extended B - Components of motion

x

y

d

y

d = x2 + y2 Magnitude of a 2D displacement

If we know the time interval t between snapshots, we can find the velocity of the ball simply by dividing the displacements shown above by t. The proportions of our triangle will not change.

vx

vy v

vy

Thus

v = vx2 + vy

2 Magnitude of a

2D velocity

Each triangle gets a good name:

displacement triangle

velocity triangle

We call the vx the horizontal component of the velocity.

Topic 1.3 Extended B - Components of motion

vx

vy v

vy

horizontal component

We call the vy the vertical component of the velocity.

vertical

component

From trigonometry we know there is a relationship between the sides of a triangle, and the angle :

opphyp

adjhyp

oppadj

hypotenuse

adjacentopposite

θ

trigonometric ratios

s-o-h-c-a-h-t-o-a

v

vx = v cos θ

vy = v sin θ

v

vxvy

vsin θ = cos θ = tan θ =

vx

vy

Suppose we know the velocity of the ball is 25.0 m/s at an angle of 30° with respect to (wrt) the positive x-axis.

Topic 1.3 Extended B - Components of motion

vx

vy v

vy

What is vx the horizontal component of the velocity?

vx = v cos θ

vy = v sin θ

vx = v cos θ

vx = (25.0 m/s)cos 30°

vx = 21.7 m/s

What is vy the vertical component of the velocity?

vy = v sin θ

vy = (25.0 m/s)sin 30°

vy = 12.5 m/sFYI: You can check your results by squaring each answer, summing, and taking the square root. What should you get?

Sometimes we know the components of the velocity, and want to find the magnitude and the direction:

Topic 1.3 Extended B - Components of motion

vx

vy v

vy

Suppose vx = 30.0 m/s.

Suppose vy = 40.0 m/s.

Thenv = vx

2 + vy2

v = 302 + 402

v = 50.0 m/s magnitude

of v

oppadj

tan θ =vx

vy=40 m/s

=30 m/s

and

so that

θ = tan-1 43

= 53.1°direction

of v

Sometimes we know the formulas for the components of the velocity of a ball, and want to find the magnitude and the direction of the velocity at a particular time:

Topic 1.3 Extended B - Components of motion

Suppose vx = 30.0 (measured in m/s).Suppose vy = 40.0 - 5t (vy in m/s, t in s)Then what is the velocity at t = 2 s?

v = vx2 + vy

2

v = 302 + 302

v = 42.4 m/s magnitude

of v

oppadj

tan θ =vx

vy=30 m/s

=30 m/s

What is the direction of the ball at this instant?

so that

θ = tan-1(1) = 45.0°direction

of v

vx = 30.0 m/s

vy = 40 - 5(2)

vy = 30.0 m/s