Mechanical Engineering - Engineering Mechanics - Discussion

The range of projectile on downward inclination is always less than the range of projectile in upward inclination plane in general (you can observe the same by throwing a ball in the air).

And the angle of projection of downward inclined projectile is always greater compared to upward one. Now as per condition of the statement given in question VELOCITY AND ANGLE OF PROJECTION MUST BE SAME IN BOTH UPWARD AND DOWNWARD INCLINATION. In the hypothetical condition in which angle of inclination is same the range of downward projectile need to be increased and it becomes greater than the range of the upward projectile.

Range of projectile in a downward inclined is given by
R= (2u^2sin(a+b)cosa)/gcos^2b
Range of projectile in an upward inclined is given by
R= (2u^2sin(a-b)cosa)/gcos^2b
Where 'a' is alpha, angle made by projection with horizontal and 'b' is beta, angle made by range with horizontal.
Since the value of sin* increase from 0° to 90° and (a+b) is greater than (a-b)
The range of projectile in a downward inclined is greater than Range of the projectile in an upward inclined.

In case of upward projection, object is moving against the gravity i.e. we can say velocity has to overcome the gravity for motion, but in case of downward motion the situation is totally opposite.

As we know that horizontal component of velocity remain the same during projectile but vertical component changes in case of upward motion vertical component is decreasing (to overcome velocity) casing the resultant velocity to decrease.

This can be easily explained with diagrams.

It's quite obvious that when the ANGLE OF PROJECTION IS SAME, the range of the projectile is more when the inclination is downwards than when the inclination is upwards.

The upwards inclination already reduces the distance the body can travel when projected as compared to a downwards inclination.

Just employ imagination and diagrams. Better still, try it out on a hill.

@All.

Let me explain. Imagine you are standing on an upward inclined plane. Now when you make a projectile, you will have to overcome the peak and thus it reduces your range.

But when you are standing on a downward inclined plane, then you will be able to throw at any projectile and thus you will get a larger range.

(Imagine yourself firing an artillery gun from a mountain).

Upward acting projectile is the one when you throw the ball from ground and the downward acting projectile is when you throw the after going to certain height. Both make trajectories.

Now, distance traveled in a parallel (i.e, when downward projectile) will be more than the distance traveled when thrown from the ground (i.e, upward projectile) while in flight.

When projectile moves upward its velocity reduce because negative gravitational acceleration there for time increase to reach same height but time taken to reach the ground less due to ground. And distance (range) = v/t.

As t increases d reduce &t reduce d increase. There for distance (range) inversely proportional to time.

While moving upward gravitational force restricts the motion of the body on the other hand, while moving downward gravitational force help the body to move downward.

So with force range of projectile (body projected at a target with particular force, force for both upward and downward motion is same) increases.

As the projectile travels through the air, the horizontal component of velocity remains constant, but the vertical component of velocity changes due to the effect of gravity, which pulls the projectile downward.

This results in the projectile travelling a greater horizontal distance before it hits the ground.

In case of downward direction gravity helps to velocity weather the upward direction gravitational force opposes the moving velocity.

That's why the range of projectile on a downward inclined plane is MORE THAN the range on upward inclined plane for the same velocity of projection and angle of projection.