Kinematics: Two-Dimensional Motion

Two-Dimensional Motion Problem Solving

So let’s apply this new information to problem solving. Look at the following examples.

Example 1: Time and Range

Horizontal Motion Off a Cliff

The vector components for an object in free fall and the vector components of an object experiencing projectile motion.

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A diver running 1.8 m/s dives out horizontally from the edge of a vertical cliff and 3.0 s later reaches the water below. How high was the cliff, and how far from its base did the diver hit the water?
Given:
V not x equals one point eight meters per second; V not y equals 0 meters per second; T equals three point zero second
Find:
Vertical height (Delta y) and range (Delta x)		 Equation:
Y equals y not plus v not y times t plus one half times g times t squared

X equals x not plus v not x times t
Solution:
First, find the vertical height of the cliff:

Y equals y not plus v not y times t plus one half times g times t squared; Delta y equals zero plus one half times nine point eight times three squared; Delta y equals forty four meters

Then, find the range of the diver:

X equals x not plus v not x times t; Delta x equals one point eight meters per second times three seconds; Delta x equals five point four meters

Example 2: Time

Football Launched at an Angle

The vector components of velocity of a football at several points along its trajectory after it is kicked at an angle theta not. At the top of the parabolic path the velocity in the y direction is 0.

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A football is kicked at ground level with a speed of 18.0 m/s at an angle of 35° to the horizontal. How much later does it hit the ground?
Given:
v equals eighteen point zero meters per second , thirty five degrees; v not y equals negative v sub y equals eighteen point zero times sine of thirty five degrees equals ten point three two meters per second		 Note: Use the trig relationships to solve for the y-component of the velocity. The initial velocity and the final velocity in this case would be the same magnitude, but in opposite directions, thus the negative sign.
Find:
Time, t		 Equation:
V sub y equals v not sub y plus g times t
Solution:
Delta v equals g times t; Ten point three two meters per second minus negative ten point three two meters per second equals nine point eight times t; Twenty point six four meters per second divided by nine point eight equals nine point eight times t divided by nine point eight; T equals two point one seconds

It took 2.1 s for the projectile to travel the entire distance.