2nd Law of motion

Newton's second law of motion

Concept of linear momentum

Linear momentum is defined as the product of mass and velocity of an object.

$p = mv$

Where:

$p$ = momentum $m$ = mass (in kg) $v$ = velocity (in m/s)

The SI unit of momentum is $\text{kg} \cdot \text{m/s}$.

Newton's 2nd law statement

"The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction of the force."

Let a force $F$ act on a body of mass $m$ changing its velocity from $u$ to $v$ in time $t$.

Step 1: Change in momentum

$\text{Change in momentum} = mv - mu$

Step 2: Rate of change of momentum

$\frac{mv - mu}{t}$

Step 3: From Newton's 2nd law

$F \propto \frac{mv - mu}{t}$

Since:

$\frac{v - u}{t} = a \Rightarrow \frac{mv - mu}{t} = ma$

So we get:

$F \propto ma$

Introducing the constant of proportionality $k$:

$F = kma$

In SI units, $k = 1$, so the equation becomes:

$F = ma$

Definition of one Newton (1 N)

If $m = 1\,\text{kg}$ and $a = 1\,\text{m/s}^2$, then:

$F = ma = 1 \times 1 = 1\,\text{N}$

Therefore: 1 Newton is the force which gives a mass of 1 kg an acceleration of 1 m/s^2^.

Applications of Newton's second law of motion

1. Car Acceleration: If a driver applies the same force to two cars of different masses, the lighter car will accelerate more than the heavier one because acceleration is inversely proportional to mass. Example: A small car accelerates faster than a truck when both are pushed with the same force.
2. Tennis Racket Hitting a Ball: When a tennis racket hits a ball, the force applied determines the acceleration of the ball. The heavier the racket and the harder the swing (force), the faster the ball will accelerate. The mass of the ball also affects its acceleration. Example: A tennis player hits a ball with more force, causing the ball to accelerate and travel faster across the court.
3. Rocket Launching: The force from the rocket's engines accelerates the rocket upward. The rocket's mass and the force generated by the engines determine how quickly the rocket will accelerate as it lifts off the ground. Example: A rocket experiences high acceleration when large amounts of force are applied by the engines, overcoming its mass and gravitational pull.
4. Friction and Sliding Objects: The frictional force on an object affects its motion. When you push a box, the force required to move it depends on both the mass of the box and the friction between the box and the surface. If the friction is high, more force is needed to achieve the same acceleration. Example: A heavy box on a rough floor requires more force to move than a light box on a smooth floor.
5. Human Strength in Lifting Objects: When lifting objects, the force exerted by a person depends on the object's mass and how quickly it needs to be lifted. The greater the mass of the object or the higher the desired acceleration, the more force is needed to lift it. Example: Lifting a heavy suitcase requires more force than lifting a light backpack, even if both need to be lifted at the same speed.
6. Vehicles Braking: The deceleration of a vehicle when the brakes are applied depends on the vehicle's mass. The more massive the vehicle, the more force is needed to decelerate it at the same rate. If a car is heavy, a larger braking force is required to stop it in the same distance. Example: A bus takes longer to stop than a bicycle because the bus has a greater mass and requires more braking force to achieve the same deceleration.
7. Sports: Baseball Batting: The force applied by the bat determines how fast a baseball will accelerate when hit. The acceleration of the baseball after being hit by the bat depends on both the force of the hit and the mass of the ball. Example: A fast pitch is harder to hit with the same bat because the force needed to accelerate the ball to the same speed is greater due to its higher mass.