i. Explain clearly with examples the distinction between magnitud… - Vidyadip

Question

$i.$ Explain clearly with examples the distinction between magnitude of displacement over an interval of time and the total length of the path covered by a particle over the same interval.
$ii.$ A body starting from rest accelerates uniformly along a straight line at the rate of $10\ ms^{-2}$ for $5 s$ . It moves for $2 s$ with uniform velocity of $50\ ms^{-1}$. Then it retards uniformly and comes to rest in $3 s$ . Draw velocity$-$time graph of the body and find the total distance travelled by it.

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Answer

$i.$ Magnitude of displacement of a particle in motion force given time is the shortest distance between the initial and final positions, while total length of path or $($path length$)$ is the length of actual path traversed by the particle in the given time.
Suppose an object goes from $A$ to $C$ following the path $A B C$, in a certain time $t$
Total length of path $=A B+B C$
When an object goes on the path $A B C$, then the displacement of the object is $(\overrightarrow{A C})$. The arrow head at $A C$ shows that the object is displaced from $A$ to $C$ .

Given: $a = 10ms^{-2}, u = 0$

$t = 5$
$\therefore$ $v = 0 + 10 \times 5$
$v = 50 = ms^{-1}$
Area below $v-t $ graph gives distance travelled in the straight line
$\therefore$ Distance $=\frac{1}{2}(50)\times(10+2)=300\text{m}$

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