Question

Q.1) obtain equation v=u +a×t Q.2) obtain S=u×t+1/2×a×t^2 Q.3) obtain v^2-u^2=2as Q.4) distinguish speed and velocity Q.5) what is uniform circular motion? Write it's formula. Q.6) Define the following terms Displacement, velocity,speed, acceleration, uniform motion, non uniform motion

Answer 1

Answer:

1). We know , a=(v-u)/t

at=v-u

v=u+at

2). We know ,

Distance = average speed x time

s = t x (u+v)/2

s = t x ( u + u +at)/2

s = t x (2u +at)/2

s = t(u + 1/2at)

s = ut + 1/2at^2

3). We know , v = u + at

t = (v-u)/a  ...(i)

Substitute (i) in s = ut + 1/2 at^2

s=u(  v−u  )/a+  1/2a((   v−u  )/a)^ 2

⇒2as=2u(v−u)+(v−u)  2  

⇒2as=2uv−2u^2  −v^2 −2uv−u^ 2  

⇒2as=v ^2  − u^2  

⇒v^2 =u^2 +2as

4). Speed is the distance covered per unit time . It is a scalar quantity i.e. it does not depend upon direction.

Velocity is the displacement of a body per unit time . It is a vector quantity i.e it depends upon the direction of the movement.

5). Uniform circular motion is the accelerated motion of a body with  constant speed and variable velocity.

6). Displacement is the shortest distance covered by a body to reach one place to another

Acceleration is the change in speed per unit time.

Answer 2

1.

Consider a particle is at position $\sf{x=0}$ with initial velocity $\sf{v=u}$ at time $\sf{t=0}$ and has displacement $\sf{x=s}$ with a velocity $\sf{v}$ at a time $\sf{t}$ moving with constant acceleration $\sf{a.}$

We know acceleration is first derivative of velocity.

$\longrightarrow\sf{a=\dfrac{dv}{dt}}$

$\longrightarrow\sf{dv=a\ dt}$

$\displaystyle\longrightarrow\sf{\int\limits_u^vdv=\int\limits_o^ta\ dt}$

$\displaystyle\longrightarrow\sf{\int\limits_u^vdv=a\int\limits_o^tdt}$

$\displaystyle\longrightarrow\sf{\big[v\big]_u^v=a\big[t\big]_o^t}$

$\displaystyle\longrightarrow\sf{v-u=a(t-0)}$

$\displaystyle\longrightarrow\sf{v-u=at}$

$\displaystyle\longrightarrow\underline{\underline{\sf{v=u+at}}}$

2.

From first equation of motion,

$\displaystyle\longrightarrow\sf{\dfrac{dx}{dt}=u+at}$

$\displaystyle\longrightarrow\sf{dx=(u+at)\ dt}$

$\displaystyle\longrightarrow\sf{\int\limits_0^sdx=\int\limits_0^t(u+at)\ dt}$

$\displaystyle\longrightarrow\sf{\int\limits_0^sdx=\int\limits_0^tu\ dt+\int\limits_0^tat\ dt}$

$\displaystyle\longrightarrow\sf{\int\limits_0^sdx=u\int\limits_0^tdt+a\int\limits_0^tt\ dt}$

$\displaystyle\longrightarrow\sf{\big[x\big]_0^s=u\big[t\big]_0^t+a\left[\dfrac{t^2}{2}\right]_0^t}$

$\displaystyle\longrightarrow\sf{s-0=u(t-0)+\dfrac{a}{2}\left(t^2-0^2\right)}$

$\displaystyle\longrightarrow\underline{\underline{\sf{s=ut+\dfrac{1}{2}\,at^2}}}$

3.

We see that,

$\displaystyle\longrightarrow\sf{a=\dfrac{dv}{dt}}$

$\displaystyle\longrightarrow\sf{a=\dfrac{dv}{dx}\cdot\dfrac{dx}{dt}}$

Since $\sf{\dfrac{dx}{dt}=v,}$

$\displaystyle\longrightarrow\sf{a=v\,\dfrac{dv}{dx}}$

$\displaystyle\longrightarrow\sf{v\ dv=a\ dx}$

$\displaystyle\longrightarrow\sf{\int\limits_u^vv\ dv=\int\limits_0^sa\ dx}$

$\displaystyle\longrightarrow\sf{\int\limits_u^vv\ dv=a\int\limits_0^sdx}$

$\displaystyle\longrightarrow\sf{\left[\dfrac{v^2}{2}\right]_u^v=a\big[x\big]_0^s}$

$\displaystyle\longrightarrow\sf{\dfrac{v^2-u^2}{2}=a(s-0)}$

$\displaystyle\longrightarrow\sf{\dfrac{v^2-u^2}{2}=as}$

$\displaystyle\longrightarrow\underline{\underline{\sf{v^2-u^2=2as}}}$

4.

Speed of a body is the distance covered by it in a path to travel from one point to another in unit time.

• Speed = Distance travelled / Time

• Speed is a scalar quantity, i.e., it has only magnitude but no direction.

• Speed is always greater than or equal to zero.

• SI unit of speed is $\sf{m\,s^{-1}.}$

• Dimension = $\sf{\left[L\,T^{-1}\right].}$

Velocity of a body is change in its position from one point to another in unit time along the straight line path between the two points.

• Velocity = Displacement / Time

• Velocity is a vector quantity, i.e., it has magnitude as well as direction.

• Velocity can be positive, negative or even zero.

• SI unit of velocity is $\sf{m\,s^{-1}.}$

• Dimension = $\sf{\left[L\,T^{-1}\right].}$

5.

Uniform circular motion is defined as the motion of a body along a circular path with uniform speed.

6.

Displacement:- Straight line path distance between initial and final points.

Velocity:- Rate of change of position of a body, or displacement of a body in unit time.

Speed:- Distance covered by a body in unit time.

Acceleration:- Rate of change of velocity of a body.

Uniform motion:- Motion of a body with constant speed.

Non - uniform motion:- Motion of a body with a varying speed.