(a) Derive second equation of motion S=u t+frac{1}{2} a t^{2} graphically where symbols have their u

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7. MOTION

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$(a)$ Derive second equation of motion $S=u t+\frac{1}{2} a t^{2}$ graphically where the symbols have their usual meanings.

$(b)$ A car accelerates uniformly from $18\, km h ^{-1}$ to $36\, km h^{-1}$ in $5$ seconds. Calculate the acceleration and the distance covered by the car in that time.

Option A

Option B

Option C

Option D

Solution

$(a)$ The area under the graph is the area of the rectangle $OACD$ plus the area of the triangle $ABC$ on top of it as shown in figure. The rectangle has a height $u$ and a length $t$. This area is the distance travelled by the object.

Hence, $S=u t+\frac{1}{2} \times t \times(v-u)$ $….(1)$

But from the expression $v=u+a t,$ we have

at $=v-u$. Substituting in equation $(1),$ we have

$S=u t+\frac{1}{2}(a t) t=u t+\frac{1}{2} a t^{2}$ $…(2)$

$(b)$ Given $u=18 km h ^{-1}=5 m s ^{-1}$

$v=36 km h ^{-1}=10 m s ^{-1}, t=5 s$

Using $v=u+a t,$ we have

$a=v-\frac{u}{t}=\frac{(10-5)}{5}=1 m s ^{-2}$

Also,

$S=u t+\frac{1}{2} a t^{2}=5 \times 5+\frac{1}{2} \times 1 \times(5)^{2}$

$=37.5 m$

Standard 9

Science

Under what conditions of motion, the distance covered by a body is equal to the magnitude of its displacement ?

easy

List two differences in tabular form between speed and velocity. When is a body said to have :
$(a)$ uniform velocity and $(b)$ variable velocity. How is the average velocity of a body is calculated when its velocity changes at a non$-$uniform rate ?

medium

If the displacement of a body is proportional to the time elapsed, what type of motion does the body possess ?

easy

An object is moving along a straight line with uniform acceleration. The following table gives the velocity of the object at various instants of time

Time $(s)$ $0$ $1$ $2$ $3$ $4$ $5$ $6$

Velocity $\left( m s ^{-1}\right)$ $2$ $4$ $6$ $8$ $10$ $12$ $14$

Plot the graph.

From the graph.

$(i)$ Find the velocity of the object at the end of $2.5 sec$

$(ii)$ Calculate the acceleration.

$(iii)$ Calculate' the distance covered in the last $4$ sec.

medium

A motor bike running at $5\, m s ^{-1}$, picks up a velocity of $30\, m s ^{-1}$ in $5\, s$. Calculate $(i)$ acceleration $(ii)$ distance covered during acceleration.

medium

Time $(s)$	$0$	$1$	$2$	$3$	$4$	$5$	$6$
Velocity $\left( m s ^{-1}\right)$	$2$	$4$	$6$	$8$	$10$	$12$	$14$

$(a)$ Derive second equation of motion $S=u t+\frac{1}{2} a t^{2}$ graphically where the symbols have their usual meanings. $(b)$ A car accelerates uniformly from $18\, km h ^{-1}$ to $36\, km h^{-1}$ in $5$ seconds. Calculate the acceleration and the distance covered by the car in that time.

Solution

Similar Questions

Under what conditions of motion, the distance covered by a body is equal to the magnitude of its displacement ?

List two differences in tabular form between speed and velocity. When is a body said to have : $(a)$ uniform velocity and $(b)$ variable velocity. How is the average velocity of a body is calculated when its velocity changes at a non$-$uniform rate ?

If the displacement of a body is proportional to the time elapsed, what type of motion does the body possess ?

A motor bike running at $5\, m s ^{-1}$, picks up a velocity of $30\, m s ^{-1}$ in $5\, s$. Calculate $(i)$ acceleration $(ii)$ distance covered during acceleration.

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$(a)$ Derive second equation of motion $S=u t+\frac{1}{2} a t^{2}$ graphically where the symbols have their usual meanings.

$(b)$ A car accelerates uniformly from $18\, km h ^{-1}$ to $36\, km h^{-1}$ in $5$ seconds. Calculate the acceleration and the distance covered by the car in that time.

List two differences in tabular form between speed and velocity. When is a body said to have :
$(a)$ uniform velocity and $(b)$ variable velocity. How is the average velocity of a body is calculated when its velocity changes at a non$-$uniform rate ?