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Horizontal Motion Unaffected by Gravity
by Ron Kurtus (revised 14 February 2011)
The horizontal velocity of an object is unaffected by the force of gravity for relatively short displacements. This means that the horizontal velocity is constant, while the vertical velocity is accelerating. The reason is because perpendicular vectors act independently of each other.
An object projected at an angle to gravity can be broken into its horizontal and vertical components. This horizontal velocity component is also unaffected by the force of gravity. Again, this is for short displacements.
However, for greater displacements, the curvature of the Earth comes into play, and the angle of the force of gravity changes with displacement. This change in angle affects the motion of the object and its velocity is no longer independent.
Questions you may have include:
- How is horizontal motion independent?
- What happens to motion projected at an angle?
- What happens a greater displacements?
This lesson will answer those questions.
Useful tools: Metric-English Conversion | Scientific Calculator.
Horizontal motion independent
For relatively short displacements, the Earth can be considered flat. In such a case, the force of gravity is continually perpendicular to an object moving in a horizontal direction.
Note: Displacement is a vector in a specific direction. Distance is a scalar quanity that is not necessarily in a straight line and where no direction is indicate.
(See Convention for Direction in Gravity Equations for more information.)
Horizontal velocity constant
From Newton's Law of the Conservation of Momentum, the velocity of an object moving in a given direction will remain constant provided there are no forces acting in that direction. A force perpendicular to the direction of motion may change the direction of the object but will have no effect on the velocity in the given direction.
This means that a horizontal velocity is independent of a vertical force or resulting velocity, such as caused by gravity.
(An application of this can be seen in Effect of Gravity on Sideways Motion.)
In the illustration below, the initial velocity of the object (vi) is the constant horizontal velocity (vx). The velocity caused by gravity (vg) is the velocity in the y-direction (vy). The Earth is considered flat for short displacements.
Velocity from gravity is independent of initial horizontal velocity
Vector explanation
Another explanation is the rule that perpendicular vector quantities are independent of each other. A vector is a graphical representation of a force, acceleration, velocity or displacement, giving both magnitude and direction.
Perpendicular components of velocity at an angle
In the illustration above, vx and vx are perpendicular and act independently of each other. The diagonal v is the vector sum of the velocities and represents that actual motion of the object.
You can also break a vector into its perpendicular components. The velocity v in the illustration above can be broken into its x and y components.
Magnitudes
The magnitude of a velocity vector is its speed, which is the absolute or positive value of the velocity.
The magnitudes of the vectors in the illustration above are:
sy = s*sin(a)
sx = s*cos(a)
where
- sy is the speed or absolute value of vector vy
- s is the speed for vector v
- sin(a) is the sine of angle a
- sx is the absolute value of vector vx
- cos(a) is the cosine of angle a
Also, according to the Pythagorean Theorem:
s2 = sx2 + sy2
(See Convention for Direction in Gravity Equations for more information.)
Applied to motion projected at an angle
This principle can be applied to motion when the initial velocity is projected at an angle to the ground. The motion can be broken into its horizontal and vertical components. The vertical component of the velocity is in the same plane at gravity and is thus affected by gravity. Meanwhile the horizontal component is perpendicular to gravity and is independent of that force.
Initial velocity broken into horizontal and vertical components
The in illustration above, the initial velocity −vi is broken into its components in the x and y directions. According to our direction convention, up is negative.
The vertical component of the initial velocity, −vy, acts only on the velocity toward the Earth, as seen in Velocity Equations for Objects Projected Upward. The horizontal velocity is unaffected by the vertical motion.
An example of this type of motion is seen in Effect of Gravity on an Artillery Projectile.
Situation of curved surface
The rule that perpendicular motion is unaffected by gravity is only applicable when the displacements are small enough that the Earth's surface is considered flat.
When the curvature of the Earth comes into play, the angle between the force of gravity and the motion of the object changes as the object moves. This invalidates the rule and complicates the situation. The effect of gravity now changes the horizontal velocity.
The only time the velocity remains constant is the special case of a circular orbit. In this case, the object moves at a constant velocity tangent to the circular path.
Angle between force of gravity and sideways velocity changes
(See Gravity and Newton's Cannon and Circular Planetary Orbits for examples)
Summary
For short displacements, the horizontal velocity of an object is constant and acts independently of the vertical force of gravity. Likewise, an object projected at an angle to gravity can be broken into its horizontal and vertical components. The horizontal velocity component is also unaffected by the force of gravity.
However, for greater displacements, the curvature of the Earth comes into play, and since the angle of the force of gravity changes with displacement the horizontal velocity is no longer independent of gravity.
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Resources and references
The following resources provide information on this subject:
Websites
Independence of Perpendicular Components of Motion - PhysicsClassroon.com
Acceleration due to Gravity Calculations - from Western Washington University
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
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Horizontal Motion Unaffected by Gravity