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Physics - Circular and Projectile Motion

Think about an object moving forward in a horizontal direction at a constant velocity. If a vertical force comes along, the force has no component that acts in the horizontal direction therefore it cannot change the objects’ speed. If an object is moving, the magnitude of its velocity cannot be changed by a force that is perpendicular to the direction of motion. So, what happens is the force does change the velocity of the object, but it does so by changing the objects’ direction not its speed. Change in direction is a change in velocity. Now, imagine a large object traveling through space in a straight line with no forces to change its velocity. It will travel forever in a straight line until some force comes along to change the situation. 

Now, imagine that a planet 300,000 miles away from the object is in a line that is perpendicular to the motion of the object. All of a sudden the object experiences a vertical force. This is the force of gravity exerted by the planet. The planet and the object attract each other with the force given by Newton’s Law of universal gravitation, F = G m1m2 / r2 where m1 and m2 equal the masses of the objects, r equals the distance between the object, and G equals Newton’s constant 6.7 x 1011 Newton meter squared per kilogram squared. 

The object changes direction and begins to follow a circular path orbiting the planet. The object just turned into a satellite and actually at each instant the satellite does have its original horizontal velocity, but then in the next instance before the satellite can get away the force of gravity changes its direction and bends it toward the planet. This keeps happening instant by instant. So the satellite moves in a circle around the planet. The satellite does not crash and it does not slow down because no force ever interfered with the magnitude of its velocity. It keeps traveling at the same speed with the direction that is constantly changing that is called centripetal acceleration. 

Centripetal acceleration means acceleration toward the center of a circular orbit. Centripetal acceleration is due to some force acting on the object in the direction of the center of the orbit. In the case of a satellite orbiting the earth, the force is the force of gravity on the satellite. For an object moving at a constant speed in a circular orbit, there is an important relationship among the speed, the radius of the orbit, and the centripetal acceleration. 

Rule number 9: 
For an object moving at a constant speed in a circular orbit, centripetal acceleration equals speed² divided by radius or a = v² / r. If an object is moving in a circular path, but it is not moving at a constant speed, then it is experiencing two forces. One force exerts an acceleration toward the center of the orbit. The other causes acceleration instant by instant, along a line that is tangent to the orbit. The two acceleration vectors together produce a resultant acceleration vector that points each instant somewhere between the two. The resultant vector represents the direction of acceleration at a given instant. 

Projectile Motion 

Supposed you kick a football, it ascends with a certain velocity at a certain angle. Let us say the angle is 30 degrees and the velocity is 10 meters per second. The football is a projectile and its velocity has two vector components. One is vertically upward and one is horizontal. 

The vertical component of the velocity equals 10 x cosine 30 which equals to 1.54. The horizontal component of the velocity equals 10 x cosine 60 which equals -9.52. Once the football is launch, there are no horizontal forces acting on it. The vertical force acting on the football is gravity which slows the ascending football with an acceleration of 9.8 meters per second squared. 

During the course of flight, the football’s horizontal velocity remains constant because no horizontal force acts on the football. During the course of flight, the upward vertical velocity is reduced. Ultimately, it is reduced to zero and then with the force of gravity continuing to operate. The football acquires a downward velocity. 

How long does it take until the football reaches the highest point in its path? At the instant we kick the football it had an upward velocity of 1.54 meters per second. The Earth’s gravity is causing the football to slow down. We want to know how much time before the football in terms of vertical motion comes to a stop in midair. So that its final velocity equals zero. 

The formula we need is Vfinal = Vinitial + 80. If Vfinal = 0, then Vinitial equals 1.54 meters per second. So that 0 = 1.54meter per second + -9.8 meters per second² times time. 

When we solve for time, time = 0.16 seconds. Now, substituting this time in the formula x = Vinitialt +1/2 AT², we can solve the equation for X to find how high the ball was kicked. 

A projectile is just like two different moving objects. One has an initial horizontal velocity that never changes and the other has an initial upward velocity, but it experiences the force of gravity. The upward vertical velocity is ultimately reduced to zero and then converted to a downward vertical velocity accelerated at 9.8 meters per second² until the object hits the earth. The highest point in the projectile’s path is the point at which the vertical velocity has been reduced to zero.