Two-Dimensional Motion and Vectors

CP: 6.1

Projectile Motion Assumptions

The acceleration of gravity is a constant -9.8 m/s2

The effect of air resistance is negligible

The rotation of the Earth has no effect.

Projectile motion only applies to bodies in free fall

Not in free fall

Projectiles are moving in 2 dimensions

Therefore, we need to look in two dimensions (the x-direction & y-direction) when solving projectile problems.

The motion on the y axis is independent of the motion on the x axis.

y axis

free fall motionx axis

constant velocity motion.

We will see in the next chapter, this is Newton’s First Law of Motion.

On the horizontalx = v t

On the verticalx = vit+½ at2

This leads to a parabolic path

For Example…

A cannon has a muzzle velocity of 62.3 m/s. What is its range when shot at an angle of 30.00o?

1. Draw a vector diagram, and resolve the velocity vector into rectangular components.

62.3 m/s

30o

62.3cos30

62.3

sin

30

Range ( x)

Example: A cannon has a muzzle velocity of 62.3 m/sec. What is its range when shot at an angle of 30.00o?

2. Using the y axis component, and the equations of motion for free fall, calculate the time of flight. (How long the projectile is in the air)6

2.3

sin

30

vi = 62.3sin30 = 31.15 m/sec

a = -9.8 m/sec2

y = 0

t = ?

y = vit + ½ at2

0 = 31.15t + ½(-9.8)t2

0 = (t)(31.15 – 4.9t)

t = 6.357sec

(Y axis motion only)

3. Using the time of flight, calculate how far the projectile will travel horizontally during that time.

x = vx t

x = 62.3 cos30 x 6.357 secx = 53.95 m/sec x 6.357 sec

x = 342.96 ~ 343 m

X Axis Motion Only

The maximum range of a projectile occurs at 45o.

Misconception #1

Going faster horizontally means you don’t fall as fast.

Misconception #2:

Gravity won’t act on you until you look down.

That is just so wrong!

A battleship simultaneously fires two shells at enemy ships. If the shells follow the trajectories shown, which ship gets hit first?

A B

1. A will hit first 3. Both will hit at the same time

2. B will hit first 4. Depends on the actual angles.

A golfer makes a shot to a tee as shown. Assuming he shoots at a 60.0o angle, with a velocity of 100. ft/sec what is the range (dx) to the tee? (UNITS!)

60o

75 ft

R ft

Example #2 Initial velocity vector

60o

75 ft

Find components of the initial velocity vector

100 cos 60

100100 sin 60

On the y axis

a = -32 ft/sec2

viy = 100 sin 60o

d = + 75 ft

Vertical displacement when

the ball is at the elevation of the tee

t = ?

Using our standard equations of motion…

On the y axis

a = -32 ft/sec2

viy = 100 sin 60o

= 86.6

d = + 75 ft

t = ?

d = vit + ½ at2

75 = (86.6)t + (-16)t2

-16t2 + 86.6t – 75 = 0

T = 1.08 sec. & 4.33secAs per the diagram, assume the long shot.

60o

75ft

R ft

1.08 sec

4.33 sec

On the x axis…

v = 100cos60o = 50 ft/sect = 4.33 sec

Range ( R) = vx t

= 50 ft/sec (4.33 sec) = 217 ft

Which ball spends more time in the air?

Which ball has the greater launch speed?

same

B

The time of flight depends only on the vertical component of the initial velocity. In this case, the vertical component is the same, ie—both balls reached the same height, so they will spend the same time in the air.

Since Ball A has the shorter range, the horizontal component of its initial velocity must be less than that of Ball B. So Ball A has a smaller launching speed.

Which ball spends more time in the air?

Which ball has the greater launch speed?

Ball B spends more time in the air.

Ball B has the greater launch speed.

Ball B spends more time in the air.

Again, the time of flight depends only on the vertical component of the initial velocity.

Ball B goes higher, so it must spend more time in the air.

Ball B has the greater launch speed.

Both balls have the same range. We know that 45o gives maximum range for a given speed.

Equivalently, 45o is the angle required for the smallest launch speed to achieve a given range.

Ball B has the greater launch speed.The closer the launch angle is to 45o, the closer the launch speed is to this smallest speed. The launching angles of both balls is greater than 45o. But, notice that Ball A’s launch angle is closer to 45o than Ball B’s. So Ball A has the smaller launch speed of the two.