Question

if a body Travels with a uniform acceleration a1 for tome t1 and another uniform acceleration a2 for time t2, then average acceleration is given derive also

Answer 1

Equations of motion for a uniform accelerated body changing its velocity u to v in time t, with an acceleration a and covering a displacement s in the time.

v = u + at

s = ut + at²/2

v² = u² + 2as

Solution

Let the initial velocity of the body be u,

The body travels with acceleration a₁ for time t₁

Final Velocity for this time period is,

v = u + a₁t₁

Now, The body accelerates with a₂ for time t₂

The velocity after this time interval is given,

w = v + a₂t₂

w = u + a₁t₁ + a₂t₂

Average acceleration :

It is the change in velocity over the time interval.

If velocity changes from a to b over the time interval t,

$\frac{a - b}{t} = avg \: acceleration$

Now Average acceleration is the change in velocity over the entire time interval.

Time t = t₁+ t₂

Initial velocity = u

Final Velocity = w

So, Average acceleration is,

$= \frac{w - u}{t} \\ \\ = \dfrac{u + a_1t_1+ a_2t_2 - u}{t} \\ \\ = \dfrac{ a_1t_1+ a_2t_2 }{t} \\ \\ = \dfrac{ a_1t_1+ a_2t_2 }{t_1+ t_2}$

Therefore, The average acceleration of the body is

$\dfrac{ a_1t_1+ a_2t_2 }{t_1+ t_2} \:$

Answer 2

Answer:

Equations of motion for a uniform accelerated body changing its velocity u to v in time t, with an acceleration a and covering a displacement s in the time.

v = u + at

s = ut + at²/2

v² = u² + 2as

Solution

Let the initial velocity of the body be u,

The body travels with acceleration a₁ for time t₁

Final Velocity for this time period is,

v = u + a₁t₁

Now, The body accelerates with a₂ for time t₂

The velocity after this time interval is given,

w = v + a₂t₂

w = u + a₁t₁ + a₂t₂

Average acceleration :

It is the change in velocity over the time interval.

If velocity changes from a to b over the time interval t,

\frac{a - b}{t} = avg \: acceleration

t

a−b

=avgacceleration

Now Average acceleration is the change in velocity over the entire time interval.

Time t = t₁+ t₂

Initial velocity = u

Final Velocity = w

So, Average acceleration is,

\begin{gathered}= \frac{w - u}{t} \\ \\ = \dfrac{u + a_1t_1+ a_2t_2 - u}{t} \\ \\ = \dfrac{ a_1t_1+ a_2t_2 }{t} \\ \\ = \dfrac{ a_1t_1+ a_2t_2 }{t_1+ t_2}\end{gathered}

=

t

w−u

=

t

u+a

1

t

1

+a

2

t

2

−u

=

t

a

1

t

1

+a

2

t

2

=

t

1

+t

2

a

1

t

1

+a

2

t

2

Therefore, The average acceleration of the body is

\dfrac{ a_1t_1+ a_2t_2 }{t_1+ t_2} \:

t

1

+t

2

a

1

t

1

+a

2

t

2