Force Formula: Momentum, Velocity & Derivation

Force is the power that makes things move, stop, change shape, or do anything different. It's like a helper that tells things how to behave when you give them a little push or pull. 

• Imagine one playing with a toy car. When one pushes the car, it starts moving. What's making it move? That's force! 
• Force is like a little invisible push or pull that can make things change their motion.
• For example, if one wants to kick a ball, your leg applies force to the ball. 
• If one wants to stop a rolling ball, you use your hand to apply force in the opposite direction.

Key Terms: Acceleration, Mass, Momentum, Velocity, inertia, centrifugal force, Centripetal force

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What is Force?

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Force is defined as an interaction that causes an object to accelerate, change its velocity, or deform. 

• It is a vector quantity, which means it has both magnitude (how strong the force is) and direction (which way the force is applied). 
• Force is typically measured in units of newtons (N) in the International System of Units (SI). 
• It can be the result of various physical interactions, such as contact between objects, gravitational attraction, and electromagnetic fields.

Force

Read More: Force of Attraction Formula

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Force Formula

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Force is a vector quantity, which means it has a magnitude as well as a direction. According to Newton's second law, force is defined as the "product of mass and acceleration."

Force = Mass × Acceleration

Force Formula

Example- A car with a mass of 1200 kg is accelerating at a rate of 5 m/s². Calculate the force applied to the car.

Solution. Mass= 1200 Kg: Acceleration = 5 m/s²

Force = m x a

= 1200 x 5 

= 6000 Newton

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Force Formula with Velocity

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The acceleration or velocity change formula is derived from the force formula.

Acceleration = Velocity × Time

Therefore force formula can be written as 

F = mv/t

Example- A baseball with a mass of 0.145 kg is pitched with an initial velocity of 35 m/s. If the ball experiences a constant deceleration of 8 m/s² due to air resistance, what is the net force acting on the baseball during its deceleration?

Solution. Mass of the baseball (m) = 0.145 kgInitial velocity (u) = 35 m/sDeceleration (a) = – 8 m/s²

v² = u² + 2as

v² = (35 m/s)² + 2 × (-8 m/s²)  × 1

v² = 1225 m²/s² – 16 m²/s²

v² = 1209 m²/s²

v ≈ 34.77 m/s

Δv = v – u

Δv = 34.77 m/s – 35 m/s

Δv ≈ – 0.23 m/s

Force (F) = Mass (m) × Acceleration (a)

Δt = Δv / a

Δt = (-0.23 m/s) / (-8 m/s²)

Δt ≈ 0.02875 s

F = m × a

F = 0.145 kg x – 8 m/s²

F ≈ -1.16 N

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Force Formula in Terms of Momentum

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The formula for inertia is p = mv which is also known as momentum.

So, Force can be defined as the speed at which momentum changes.

F = p/t = dp/dt

Example- A tennis ball with a mass of 0.06 kg is initially at rest. A player hits the ball, giving it a constant force that results in a change in velocity from 0 m/s to 20 m/s in a time of 0.02 seconds. Calculate the force applied to the tennis ball during the hit.

Solution- Mass of the tennis ball (m) = 0.06 kg Initial velocity (u) = 0 m/s, Final velocity (v) = 20 m/s, Time (Δt) = 0.02 seconds

Initial momentum (p_initial) = mass × initial velocity = 0.06 kg × 0 m/s = 0 kg m/s

Final momentum (p_final) = mass × final velocity = 0.06 kg × 20 m/s = 1.2 kg m/s

Now, let's calculate the change in momentum (Δp):

Δp = p_final - p_initial = 1.2 kg m/s - 0 kg m/s = 1.2 kg m/s

Force (F) = Change in Momentum (Δp) / Time (Δt)

F = 1.2 kg m/s / 0.02 s

F = 60 N

Read More: Gravity Waves

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Unit of Force Formula

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Force is defined or quantified as the product of mass and acceleration, according to Newton's second law of motion.

SI Unit

The unit of force in the International System of Units (SI) is Newton, abbreviated as "N." The formula for force in SI units is:

F = m ⋅ a

where:

• F represents the force in newtons (N)
• m represents the mass of an object in kilograms (kg)
• a represents the acceleration of the object in metres per second squared (m/s²)

This formula states that force is equal to the product of an object's mass and its acceleration. In simpler terms, force is what causes an object to accelerate, decelerate, or change its state of motion.

FPS System

In the FPS (Foot-Pound-Second) system of units, the unit of force is the pound-force, often abbreviated as lbf or lbF. This is the force required to accelerate a one-pound mass by one foot per second squared.

In equation form:

1 pound-force (lbf) = 1 pound-mass (lb) × 1 foot per second squared (ft/s²)

Please note that the FPS system is not as commonly used as other systems like the SI (International System of Units) or the CGS (Centimeter-Gram-Second) system, and its usage has been largely supplanted by these other systems in scientific and engineering contexts.

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Derivation of Force Formula

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Newton's second law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, it is expressed as:

F = m * a

Where:

• F is the force applied to the object (in newtons)
• m is the mass of the object (in kilograms)
• a is the acceleration of the object (in metres per second squared)

Now, let's derive this formula using basic principles.

Start with the definition of force:

Force (F) is the rate of change of momentum.

F = Δp / Δt

Momentum (p) is the product of mass and velocity:

p = m * v

Substitute momentum into the force definition:

F = Δ(m * v) / Δt

Apply the product rule of differentiation to the momentum:

F = m * Δv / Δt + v * Δm / Δt

The change in velocity (Δv / Δt) is the definition of acceleration (a):

F = m * a + v * Δm / Δt

In many cases, the change in mass (Δm / Δt) is negligible compared to the mass itself, especially for non-relativistic speeds. So, we can assume that the mass remains constant, and the change in mass term can be ignored:

F = m * a

This is Newton's second law of motion, which relates force, mass, and acceleration. This law forms the basis for understanding and calculating forces in various physical scenarios.

Also Read:

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Things to Remember

• Force is a vector quantity causing acceleration or deformation.
• Newton's second law: Force (F) = mass (m) × acceleration (a).
• Units: SI (Newton, N), CGS (dyne, dyn), FPS (pound-force, lbf).
• Gravity's force: F = m × g (acceleration due to gravity).
• Contact vs. non-contact forces (e.g., friction vs. gravity).
• Force diagrams show forces on an object.
• Net force accelerates; equilibrium has zero net force.

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