Newton Second Law of Motion and Equation of Motion


 
 
Concept Explanation
 

Newton Second Law of Motion and Equation of Motion

The magnitude of the net force acting on a body is proportional to the product of the mass of the body and its acceleration.The direction of the force is same as that of the acceleration.

F:=:kma                 ...............(i)

Here, k is a constant, m = mass and a = acceleration 

In terms of momentum second law can be stated as:

The rate of change of momentum of an object is proportional to the net force applied on the object.The direction of the change of momentum is the same as the direction of the net  force.

Let us see how the two ways of stating the second law are equivalent.Suppose a object of mass m is moving along a straight line with a constant acceleration a. Also, suppose its velocity at time  t_1 is v_1, which changes to  v_2  at time  t_2.

The linear momentum at time t_1 is  p_1:=:mv_1:and :that :at :time: t_2:is :p_2:=:mv_2.

Here, p = momentum ,t= time  and  v = velocity

The rate of change of momentum is  frac{p_2-p_1}{t_2-t_1}   

According to the second law,

                                                         frac{p_2-p_1}{t_2-t_1}:alpha :F

    or                                                  F:=:kfrac{p_2-p_1}{t_2-t_1}          (where k is a constant)     

   or                                                  F:=:kfrac{mv_2-mv_1}{t_2-t_1}:=:km[frac{v_2-v_1}{t_2-t_1}]

  or                                                   F:=:kma                            ....................(ii)

Equation (ii) is same as that of equation (i)

Equation of motion:

1. First equation of motion:    v::=::u;+;at

This equation is a relation between initial velocity, final velocity and acceleration.In lots of problems, the object starts motion from the rest with zero initial velocity. In such cases if we have knowledge about two kinematics quantities we can easily find the third quantity.

2. Second equation of motion: S ;=:ut;+;frac{1}{2}at^2

This equation gives relation between distance, initial velocity, time and acceleration.

3. Third equation of motion:   v^2;=;u^2:+:2aS

This equation give the relation between final velocityinitial velocity, acceleration and distance.

    Application of second law of motion:

1. A cricket palyer moves his hands backward while catching a fast cricket ball.

2. During athletics meet, athletes doing high jump and long jump land on foam or a heap od sand to decrease the force on the body and the landing is comfortable.

3.  When we kick a ball we apply some force in it, and in a specific direction. If applied force is more than the ball will cover a  longer distance and if the applied force is less than ball wiil cover a smaller distance.

4. Pushing a empty cart is easy than pushing a loaded cart , this is because if the mass of the cart is less than force required to pull the cart will also be less.         

Q 1. A 650 kg rocket is to be speed up from 440 metres per second to 520 meters per second in outer space. If the thrust of the engine is 1200 N, for how long must the engine be fired?

Solution:   The change in the momentum of the rocket    Delta p:=:mv:-:mu            

                           = (650) (520) - (650) (440)  =  52000 kg m/s.

                  This must be equal to the impulse, so FDelta t = (1200 N) (Delta t)  =  52000 kg m/s

                  therefore     Delta t  =   43 s

Q 2.  A boy of mass 58 kg jumps with a horizontal velocity of 3 m/s onto a stationary skateboard of mass 2 kg. What is his velocity as he moves off the skateboard ?

Solution: Assume there is zero unbalanced horizontal force in the horizontal direction and that left to right is the positive (+) direction.

                Momentum before interaction  =   m_1u_1+m_2u_2:=58times3:+;2times0

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Sample Questions
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Question : 1

A wooden block, initially at rest, can be displaced by 15m in two seconds by applying a force of 150 N. The mass of the block is _____________________

Right Option : D
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Question : 2

 The second law of motion gives the measure of which of the following?

Right Option : A
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Question : 3

A body of mass 4 kg moving on a horizontal surface with an initial velocity of 6 ms^-^1 comes to rest after 3 seconds. If one wants to keep the body moving on the same surface with the velocity of 6 ms^-^1, the force required is

Right Option : C
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