Projectile Motion

In kinematics, the motion of a particle in a straight line. We distinguished between uniform motion, such as a billiard ball moving horizontally in a constant velocity; and uniformly accelerated motion such as a mango falling vertically under the influence of gravity. Now we are going to consider objects moving in a nonlinear motion or in a curved path. When you kicked a soccer ball with a certain angle, it follows a curved path. The path is known as a trajectory. The curved path motion is a combination of a uniform horizontal motion and a uniformly accelerated vertical motion.  Take note that the velocity of the kicked ball at any instant has two components of motion. The horizontal motion does not affect the vertical motion. This motion is called projectile motion.

We are fond of playing or watching a volleyball, football or a basketball match. The ball thrown or kicked in air is an example of a projectile. A cannonball shot from a cannon, a satellite orbiting the earth are also projectiles. 

To avoid confusion on projectile motion, let us treat the horizontal component and the vertical component of the motion separately.  If friction is negligible, the horizontal component of a projectile  motion is similar to a  ball moving a straight line in a constant velocity.

The vertical component of the projectile motion is similar to a ball moving as a free falling body. The velocity increases with time.

If we combine these components we will obtain a projectile motion. A projectile is simply a combination of a uniform horizontal motion and a uniformly accelerated vertical motion. The only force acting in the motion is the weight. The path of the projectile motion is known as the trajectory. When you kicked a ball with a certain angle and it landed to a certain point, the distance between the kick off point and the landing point is what you called the range.

For example, you kicked a soccer ball with an initial velocity of 42 m/s, 37⁰ from the horizontal. What is the maximum height the ball could reach? How long it will take for the ball to reach the maximum height? How long is the ball in air? What is the range?

First you need to get the x and y component of the velocity, v_o,

v_{ox = Vo}cosθ

v_{ox =} 42m/s cos 37⁰

v_{ox =}  33.54 m/s

v_{oy =} v_osinθ

v_{oy =} 42 m/s sin 37⁰

v_oy =  25.28 m/s

To get the height, we are going to use the equation

h = (v_1y ² - v_oy²)/ -2g

Remember that at the maximum height the velocity, v₁ is zero. Therefore

h = (v_1y ² - v_oy²)/ -2g

h =  - v_oy²/ -2g

h = (25.28m/s)² /-2 (9.8m/s²)

h = 32.6 m/s

To get the time to reach the maximum height we are going to use the equation

t = (v_1y  - v_oy)/ -g

At the maximum height, the velocity v₁ is zero. Therefore

t =  - v_oy/ -g

t =-25.28/-9.8 m/s²

t = 2.58s

The time the ball is in the air is

T = 2t

T = 2 (2.58s)

T = 5.16s

To get the range we are going to use the equation

R = v_ocosθT

R = (42 cos 30)(5.16s)

R = 173.03 m

Practical Applications:

If you are going to fire a gun horizontally and drop a bullet vertical with the same height with the gun, at the same time, which bullet will hit the ground first? Why do we need to avoid indiscriminate firing?

Predict, Observe, Explain

 Put a coin at a smooth table which is slightly hanged at the edge. Then put another coin near the overhanging coin. Flick the second coin across the table so that it strikes the overhanging coin and both coins now will fall on the floor. Which coin hits the floor first?