# Force and Motion

Force and Motion

Force is an external agency that changes or tends to change the position of a body from rest to motion or motion to rest.

### Rest and Motion

A body is said to be at rest if it does not change it position with respect to surrounding.

A body is said to be at motion if it changes its position with respect to surrounding.

### Uniform and non-uniform motion

If a body covers equal distance in equal interval of time, then the body is said to be at uniform motion.

If a body covers unequal distance in equal interval of time, then the body is said to be at non-uniform motion.

### Scalar and Vector Quantity

Those physical quantities which have magnitude but no direction is known as scalar quantity. Its value is always positive. They don’t follow vector law of addition.

Examples: Speed, distance, mass etc.

Those physical quantities which have both magnitude and direction are known as vector quantity. Its value may be positive, negative or zero. They follow vector law of addition.

Example: Velocity, displacement, weight etc.

### Distance

The length of path travelled by a body is known as distance. Its SI unit is meter. It is scalar quantity. Its value is always positive.

### Displacement

The length of path travelled by a body in particular direction is known as displacement. Its SI unit is meter. It is vector quantity. Its value may be positive, negative or zero.

### Speed

The distance travelled by a body per unit time is called speed. It is scalar quantity.

Mathematically, \(speed= \frac{distance}{time}\)

Its SI unit is m/s.

### Velocity

The displacement travelled by a body per unit time is called velocity. It is vector quantity.

Mathematically, \(velocity= \frac{displacement}{time}\)

Its SI unit is m/s.

### Average velocity

Average velocity can be defined as mean of initial velocity and final velocity for given period of time.

\(Average Velocity (V_{av})= \frac{initial velocity (u)+final velocity(v)}{2}\)

### Uniform Velocity

If a body doesn’t change its velocity, then the body is said to be at uniform velocity.

### Non-uniform velocity

If a body changes its velocity, then the body is said to be at non uniform velocity.

### Acceleration

Acceleration can be defined as rate of change in velocity. Its SI unit is m/s^{2}. It is vector quantity.

Mathematically,

\(Acceleration= \frac{final velocity(v) – initial velocity(u)}{time taken(t)}\)

Negative acceleration is known as retardation.

### Relative velocity

The velocity of a body with respect to another body is known as relative velocity.

Relative velocity (V_{ab})=V_{a}-V_{b} (If body A and B are moving in same direction)

Relative velocity (V_{ab})=V_{a}+V_{b} (If body A and B are moving in opposite direction)

### Equations of Motion in Straight Line

Let us consider a body is moving with initial velocity ‘u’ and reaches final velocity ‘v’ in time ‘t’. If ‘a’ be the acceleration and ‘s’ be the distance covered by a body then,

We know that,

\(a=\frac{v-u}{t}\)

or,at=v-u

or at+u=v

So, v=u+at…………(1)

We know that,

Distance= Average velocity × time(v=d/t)

\(s=\frac{u+v}{2} × t\)

\(s=\frac{u+u+at}{2} × t\)

\(s=\frac{2u+at}{2} × t\)

\(s=(\frac{2u}{2} + \frac{at}{2})× t\)

\(s=(u+\frac{at}{2}) × t\)

\(s=(ut+\frac{1}{2}) × t\)

s= ut+½at^{2}

We know that,

v=u+at

Squaring on both sides,

v^{2}=(u+at)^{2}

v^{2}=u^{2}+2 u at + (at)^{2}

v^{2}=u^{2}+2u at +a^{2}t^{2}

v^{2}=u^{2}+2a(ut+½at^{2})

v^{2}=u^{2}+2as(s=ut+½at^{2})

### Give reason.

**Distance, speed, mass, etc are scalar quantities.**

Distance, speed, mass, etc are scalar quantities because they have magnitude but no direction.

**Displacement, velocity, weight, etc are vector quantities.**

Displacement, velocity, weight, etc are vector quantities because they have both magnitude and direction.

**Acceleration of a body with uniform velocity is zero.**

We know that,

Acceleration=final velocity-initial velocity/2

Since, final velocity and initial velocity are same as body is with uniform velocity.

So, v-u=0; a=0 m/s^{2}Thus, acceleration of a body with uniform velocity is zero.

**It is possible to have zero displacement but non zero distance travelled.**

In case of a body moving in circular track after one complete revolution, the distance travelled by a body is 2πr but zero displacement.

**Bus A finds bus B in rest if they are moving with same velocity in same direction.**

Bus A finds bus B in rest if they are moving with same velocity in same direction because the relative velocity is zero.

As, V_{ab}=V_{a}-V_{b}Since, V_{a}=V_{b}so, V_{ab}=0m/s

**Rest and motion are relative term.**

Rest and motion are relative term because they both are defined with respect to their surroundings. The same body might be in rest with one observer but might be in motion with respect to another observer.

### Numerical Problems:

**A motorbike covers a distance of 1.5km in a straight road in 2 minutes. Find the velocity of the motorbike.**

Here,

displacement(s)=1.5 km=1.5×1000m=1500m

Time taken(t)=2 min=2×60 sec=120 sec

Velocity(v)=?

We know that,

Velocity(v)=displacement(s)/time(t)

=1500/120

=12.5m/s

**A car starts from rest and gains a velocity of 20m/s in 10 s. Calculate acceleration of the body.**

Here,

Initial velocity(u)=0m/s (Since, car starts from rest)

Final velocity(v)=20m/s

Time taken(t)=10 s

Acceleration(a)=?

We know that,

a=v-u/t

=20-0/10

=20/10

=2m/s^{2}

**A bus starts from rest and attains an acceleration of 2m/s**^{2}after 10 seconds. Find the distance covered by the bus in that time.

Here,

Initial velocity(u)=0m/s (Since, car starts from rest)

Time taken(t)=10 s

Acceleration(a)=2m/s^{2}

Distance(s)=?

We know that,

s=ut+½ at^{2}

=0×10+½×2 ×10^{2}

=0+½×2 ×100

=0+100

=100m

**A car moving along a straight highway at a speed of 144 km/hr is brought to a stop within a distance of 200m**.**What is the retardation of the car?****How long does it take for the car to stop?**

Here,

Initial velocity(u)=144km/hr=144km/1hr =144×1000/60×60=40m/s

Final velocity(v)=0m/s (As car came to rest)

Distance(s)=200m

Acceleration(a)=?

Time taken(t)=?

We know that,

v^{2}=u^{2}+2as

or, 0^{2}=40^{2}+2×a×200

or,0=1600+400a

or,0-1600=400a

or, -1600=400a

or, -1600/400=a

∴a=-4m/s^{2}

(-) sign indicates retardation.

Also,

v=u+at

or, 0=40+(-4)×t

or, 0=40-4t

or, 4t=40

or, t=40/4

∴ t=10 sec

**Vehicle A is moving with velocity 25m/s. Another vehicle is moving with velocity 10m/s. Find the relative velocity of A with respect to B if,****They are moving in same direction.****They are moving in opposite direction.**

Here,

Velocity of A(Va)=25m/s

Velocity of B(Vb)=10m/s

Now,

__When they are moving in same direction__

V_{ab}=V_{a}-V_{b}=25-10=15m/s

__When they are moving in opposite direction__

V_{ab}=V_{a}+V_{b}=25+10=35m/s