Projectile Motion Down an Inclined Plane

Any object which is thrown into space and the only force acting upon it is gravity then that object is called Projectile and the motion which the body undergoes is called Projectile motion. The path undertaken by the projectile is known as Trajectory.  

We tend to assume that the object is near to the surface of the earth. When we say that only gravity acts on it, that doesn’t mean the other forces don’t, it means the effect of the other forces are minimal compared to gravity. Projectile motion is considered to be one of the most common motions in the plane. Any falling object can be considered as an example for Projectile motion.

Inclined Planes:

An inclined plane is a horizontal surface that tilts from its normal orientation. An inclined plane is also referred to as an “unbalanced” plane. The inclination of an inclined plane is equal to the angle between the normal and the plane. An inclined plane is characterized by a “critical angle” at which the plane begins to rotate to a perpendicular orientation.

Projectile motion up an inclined plane:

The projectile motion of a projectile up an inclined plane is achieved by the motion of the projectile itself. When the projectile is in free fall, the vertical component of its velocity, the speed of the projectile, is dependent on the angle of inclination, and the horizontal component of its velocity is dependent on the distance to the surface. In order to achieve the desired height and velocity, the projectile will have to accelerate through successive turns and then decelerate, as the projectile will have to travel up the inclined plane.

 Projection of a particle is upward with an initial vertical velocity up from an incline that makes an angle name α relative to the component which is horizontal t of velocity of projection. The inclined plane and the vertical component of initial velocity makes an angle θ  with the inclined plane.

Trajectory followed by a projectile motion:

The trajectory of the projectile is the path that the projectile takes during the flight from the point of release to the point of impact. For example, if you were to throw a ball into the air, it would take a trajectory, and the ball would continue to move through the air until it hit the ground. 

Time of Flight for projectile motion up an inclined plane:

Time of flight is the time taken for a projectile to travel from the point of launch to the point of impact. In the time of flight, the projectile is constantly moving forward, but the time taken for a portion of the projectile to travel is dependent on several factors, including the speed and direction of the projectile and the distance between the two points. The time of flight of a projectile can also be described mathematically as the time taken for a particle to go a certain distance in a certain direction. The formula used to calculate the time of flight of a projectile is therefore dependent on the direction and speed of the projectile.

 T = 2uSin(θ-α)

       g Cosα

Horizontal Range of flight for a projectile motion up an inclined plane:

In the case of a projectile, the range is the maximum distance from the shooter to the target. The range may be the distance from the target to the shooter. The range is often expressed in terms of height of the target above the shooter, because the height of the target is a good indicator of the range.

R = u² (1-sinα)

         gcosα

Maximum Height:

The maximum height of a projectile motion is the height of the projectile after it has been launched from its launch point at a velocity that is constant.

H = u² sin²(θ – α)

             2g

Conclusion

Under constant acceleration, when an object is thrown into space it moves along a curved path and is directed towards the center of the earth. This motion is called Projectile motion. The only force acting upon projectile motion is gravity, all the other forces are minimal so they are not considered. Projectile motion has its own path, called a trajectory. We see the use of projectile motion in our everyday lives including sports, water fountains, fireworks etc. The trajectory equation acts as a proof that all the projectile motions are parabolic in nature. We make use of the motion equations to find different parameters related to the projectile motion.