# Two-Dimensional Kinematics

An object is in equilibrium when the net force on it is zero. When in equilibrium, an object is either motionless or moves with constant velocity. According to Newton's Law, there will be no acceleration because there is no net force on it. Equilibrium also occurs when the resultant of three or more forces equals a net force of zero.

When forces to not add up to zero, the force that produces equilibrium (**equilibrant**) is the same

magnitude as the resultant but opposite in direction.

Projectile motion is the curved motion of an object that is projected into the air. The motion consists of two independent parts:

horizontal motion with a = 0 and vf = vi = vav (i.e., constant speed), and

vertical motion with a = g = 9.8 m/s2 downward.

Projectile problems can be solved easily if air friction is ignored.

Gravity is the only force that acts upon a projectile. It causes a projectile to accelerate downward. The force of gravity could never alter the horizontal velocity of an object because perpendicular components of motion are independent of each other. Vertical force does not affect horizontal motion. A vertical force acting upon a horizontally moving object will cause it to deviate from its linear path as shown below.

When vertical free fall motion is combined with constant horizontal motion, the trajectory will be that of a parabola. What type of path do you think the cannonball will follow in the following picture?

If you guessed parabolic, you're right! As the cannonball rises it undergoes deceleration (slowing down). As it falls, it undergoes acceleration due to the force of gravity. The horizontal motion is a result of its own .... Inertia.

Remember: Once an object is moving, forces will only accelerate it. Forces are not required to keep an object moving. Forces do NOT cause motion. Forces cause acceleration.

What type of path will the package dropped from a plane, flying at constant speed, follow as depicted in the following picture?

It follows a parabolic path. As the package falls, it undergoes a vertical acceleration due to the downward force of gravity. If air resistance is ignored, there is no horizontal acceleration. There would be a constant horizontal velocity. The horizontal motion is a result of what? You guessed it...inertia!!

If we launched a cannonball at three different angles: 30, 45, and 60 degrees how will they compare in trajectories? Suppose there is no air resistance and all factors except angle are held constant. Which cannonball will go the farthest? Which cannonball will go the highest? Which cannonball will hit the ground first?

Highest cannonball. The cannon set at 60 degrees.

Farthest cannonball. The cannon set at 45 degrees

First cannonball to hit ground. The cannon set at 30 degrees.

Uniform circular motion can be defined as:

**v = 2 Þr 2/ t**

The acceleration of an object in uniform circular motion always points towards the center of the circle. This center-seeking is called centripetal acceleration. This can be expressed as:

**ac =** **4 Þ2r / t2**

**Fnet = mac = mv2/r = m(4n2r/t2)**

**The "F" Word**

Is there an outward force on objects moving in a circular motion? This has been referred to as centrifugal force. It is a nonexistent force !!! What causes you to be thrown to the right side of a car when it makes a left turn? The answer is found in Newton's First Law. It is inertia !!!

If you were standing near the center of a revolving disc, would your speed be greater or less than if you were standing near the outside? See below for answer.

In the following illustration, the ball continues in a straight line (inertia) according to Newton's First Law. When something produces a centripetal force on the ball, it remains within the container moving in a circular path.

**Review Questions:**

1. Brian, who weighs 1000N, loves to play with electricity, so he hangs from two telephone lines that each make 22.5 degree angles with the horizontal. What is the tension in the lines?

1304.5 N

2. Ana fires a gun at a burglar in her home. She fires the bullet at a specific height and parallel to the ground. At the same time a bullet drops from the gun. (a) Which bullet hits the ground first? (b) If the bullet fired from the gun travels 1000 feet per second, and is fired from a height of 5 feet, how far does the bullet travel horizontally before it hits the ground?

(a) The bullets will hit the ground at the same time!

(b) 560 feet.

3. Ashley returns from her singing tour on an airplane. The airliner approaches the runway for a landing and the pilot says her horizontal airspeed is 400 miles per hour. She also hears that she is moving through a head wind of 20 miles per hour. The plane is dropping at a speed of 10 feet per second. If the plane is at a height of 3,000 feet and the start of the runway is 28 miles away (horizontally), does the pilot need to make any adjustments in descent or will the plane land at the start of the runway?

Since the runway is 28 miles away and at the current rate of descent, the plane will travel 31.7 miles, the pilot needs to slow down in order to land.

4. Victor gets on the Pro Lunar Golf Tour. He hits a golf ball on the moon with a velocity of 50 meters/second at an angle of 35 degrees above the horizontal. Remember that the acceleration of gravity is 1/6th that of Earth and that there is no air resistance on the moon.

- How far does the golf ball travel before it hits the ground?
- What is the time of flight?
- With what velocity and at what angle does the ball hit the ground?

(a) 2882 meters or 1.8 miles!

(b) 70.25 seconds

(c) The ball hits the ground at the same angle as it left the ground and at the same angle of 35 degrees.

5. Dr. Hopper, the king of snow box derbies, races down a ski slope of 37 degrees. The total mass between Dr. Hopper and his brothers is 250 kg. The coefficient of kinetic friction between the box and the snow is 0.15. How fast is he going 5 seconds after starting from rest?

24 m/s