Worked Examples-Calculation Part 1 - Motion in Straight line

Considered Formula;

$$\text {i)}\quad v = u + at$$ $$\qquad \text {Where}~~u \to Initial~velocity,\quad v \to Final~velocity \\\qquad a \to Acceleration \quad t \to Time$$ $$\text {ii)}\quad s =ut + \frac {1}{2}at^2$$ $$\qquad \text {Where}~~s \to Distance~traveled,\quad u \to Initial~velocity, \\\qquad a \to Acceleration \quad t \to Time$$ $$\text {iii)}\quad v^2 = u^2 + 2as$$ $$\qquad \text {Where}~~u \to Initial~velocity,\quad v \to Final~velocity, \\\qquad a \to Acceleration,\quad s \to Distance~traveled$$

Example 1

A car travels at 45 m/s and decreases its velocity uniformly to 20 m/s in 5 sec. find acceleration

Solution

Data given

Initial velocity (u) = 45 m/s

Final velocity (u) = 45 m/s

Time (t) = 5 s

Accelaration (a) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~v = u + at$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~20= 45 + 5a$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~5a= 20 - 45$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~5a= -25$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac {5a}{5}= \frac {-25}{5}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a = -5~m/s^2$$

The acceleration = -5 m/s²

Example 2

An electric train accelerates uniformly from rest to 180 km/h in 10 sec. Find the acceleration of the train

Solution

Data given

Initial velocity (u) = 0 m/s

$$\text {Final velocity (u)}= 180 km/h = \Biggl( \frac {180 \times 1000}{3600}\Biggr)~m/s = 50 m/s$$

Time (t) = 10 s

Accelaration (a) =?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~v = u + at$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~50= 0 + 10a$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~10a= 50 - 0$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~10a= 50$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac {10a}{10}= \frac {50}{10}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a = 5~m/s^2$$

The acceleration = 5 m/s²

Example 3

A body moving with a velocity of 30m/s is accelerated uniformly to a velocity of 50m/s in 5 s. calculate the

i)        Acceleration

ii)       Distance traveled by the body.

Solution

Data given

Initial velocity (u) = 30 m/s

Final velocity (u) = 50 m/s

Time (t) = 5 s

Accelaration (a) =?

i)        Acceleration (a) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~v = u + at$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~50= 30 + 5a$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~5a= 50 - 30$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~5a= 20$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac {5a}{5}= \frac {20}{5}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~a = 4~m/s^2$$

          The acceleration of the body= 4 m/s²

ii)        Distance (s) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~s =ut + \frac {1}{2}at^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 30(5) + \frac {1}{2}\times 4(5)^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 150 + 2(25)$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 150 + 50$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 200~m$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 200~m$$

          The distance traveled by the body = 200 m

Example 4

An object travelling at 15 m/s accelerates uniformly at 2.5 m/s² for 12 seconds.

i)        Calculate the final velocity

ii)       How far does it travel?

Solution

Data given

Initial velocity (u) = 15 m/s

Accelaration (a) =2.5 m/s²

Time (t) = 12 s

i)        Final velocity (v) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~v = u + at$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 15 + 2.5(12)$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 15 + 30$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 45~m/s$$

          The final velocity = 45 m/s

ii)        Distance (s) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~s =ut + \frac {1}{2}at^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 15(12) + \frac {1}{2}\times 2.5(12)^2$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 180 + \frac {1}{2}\times 2.5(144)$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 180 + 2.5(72)$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 180 + 180$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 360~m$$

          The distance traveled by the object = 360 m

Example 5

A body accelerates from a velocity of 15 m/s at 2 m/s² to a distance of 54m. calculate the;

i)        The final velocity

ii)       The time taken for the body to travel

Solution

Data given

Initial velocity (u) = 15 m/s

Accelaration (a) =2 m/s²

Distance (s) = 54 m

i)        Final velocity (v) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~v^2 = u^2 + 2as$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~v = \sqrt{ u^2 + 2as}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \sqrt{ 15^2 + 2(2)(54)}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \sqrt{ 225 + 216}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= \sqrt{ 441}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~= 21~m/s$$

          The final velocity = 21 m/s

ii)        Time (t) = ?

$$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~v = u + at$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~21 = 15 + 2t$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2t = 21 - 15$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~2t = 6$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\frac {2t}{2}= \frac {6}{2}$$ $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~t = 3~sec$$

          The time taken for the body to travel = 3 seconds