Question

A rocket is uniformly accelerated from rest to a speed
of 1000 m s-1 in 2.0 min. Calculate (a) the average
velocity and (b) the distance travelled​

Answer 1

Given :-

Initial velocity of the rocket = 0 m/s

Final velocity of the rocket = 1000 m/s

Time taken by the rocket = 2 min

To Find :-

The average velocity of the rocket.

The distance travelled by the rocket.

Analysis :-

Firstly, using the first equation of motion substitute the given values in the question and find the acceleration of the rocket.

Then using the second equation of motion you can find the distance travelled by the rocket.

Finally, in order to find the average velocity divide the total distance travelled by the total time taken.

Solution :-

We know that,

• v = Final velocity
• a = Acceleration
• t = Time
• u = Initial velocity

Using the formula,

$\underline{\boxed{\sf First \ equation \ of \ motion=v=u+at}}$

Given that,

Final velocity (v) = 1000 m/s

Initial velocity (u) = 0 m/s

Time (t) = 2 min = 120 sec

Substituting their values,

⇒ 1000 = 0 + a × 120

⇒ 1000 = 0 + 120a

⇒ 120a = 1000 - 0

⇒ 120a = 1000

⇒ a = 1000/120

⇒ a = 100/12

⇒ a = 8.33 m/s²

We know that,

• s = Displacement
• u = Initial velocity
• t = Time
• a = Acceleration

Using the formula,

$\underline{\boxed{\sf Second \ equation \ of \ motion=s=ut+\dfrac{1}{2} at^2 }}$

Given that,

Initial velocity (u) = 0 m/s

Time (t) = 120 sec

Acceleration (a) = 8.33 m/s²

Substituting their values,

⇒ s = 0 × 120² + 1/2 × 8.33 × 120²

⇒ s = 120²/2 × 8.33

⇒ s = 14400/2 × 8.33

⇒ s = 7200 × 8.33

⇒ s = 59976 m ≈ 60000 m

Therefore, the distance travelled by the rocket is 60000 m.

Using the formula,

$\underline{\boxed{\sf Average \ velocity= \dfrac{Total \ distance \ travelled}{Total \ time \ taken} }}$

Given that,

Distance (s) = 60000 m

Time (t) = 120 sec

Substituting their values,

⇒ 60000/120

⇒ 6000/12

⇒ 500 m/s

Therefore, the average velocity of the rocket is 500 m/s.

Answer 2

$\Large{\bf{\green{\underline{GiVeN\::}}}}$

• A rocket is uniformly accelerated from rest to a speed of 1000 m/s in 2.0 min.

$\longmapsto\:\:\bf\blue{Initial\:velocity\:(u)\:=\:0\:m/s\:}$

$\longmapsto\:\:\bf\orange{Final\:velocity\:(v)\:=\:1000\:m/s\:}$

$\longmapsto\:\:\bf\green{Time\:(t)\:=\:2\:min.\:=\:120\:s}$

$\\ \Large{\bf{\pink{\underline{To\:FiNd\::}}}}$

⑴ The average velocity.

⑵ The distance travelled by rocket.

$\\ \Large{\bf{\purple{\underline{CaLcUlAtIoN\::}}}}$

$\bf\red{We\:know\:that,}$

$\orange\bigstar\:\:{\underline{\green{\boxed{\bf{\blue{v\:=\:u\:+\:a\:t}}}}}}$

$:\implies\:\:\bf{1000\:=\:0\:+\:a\times{120}}$

$:\implies\:\:\bf{120a\:=\:1000\:}$

$:\implies\:\:\bf{a\:=\:\dfrac{1000}{120}\:}$

$:\implies\:\:\bf\pink{a\:=\:8.333\:m/s^2}$

✅ Now we calculate the distance travelled by the rocket and we use the following equation of motion.

$\purple\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S\:=\:u\:t\:+\:\dfrac{1}{2}\:a\:t^2}}}}}}$

$:\implies\:\:\bf{S\:=\:0\times{120}\:+\:\dfrac{1}{2}\times{8.333}\times{(120)^2}\:}$

$:\implies\:\:\bf{S\:=\:0\:+\:4.1665\times{14400}\:}$

$:\implies\:\:\bf{S\:=\:59997.6\:m\:\approx\:60000\:m}$

$:\implies\:\:\bf\blue{S\:=\:60\:km}$

$\\ \Large\bf\pink{Therefore,}$

⑵ The distance travelled by rocket is 60 km.

✅ To calculate average velocity, we use the following formula.

$\green\bigstar\:\:{\underline{\blue{\boxed{\bf{\purple{Average\:velocity \:=\:\dfrac{Total\:Displacement}{Total\:Time}\:}}}}}}$

$\bf\red{Where,}$

• Total displacement = 60 km = 60000 m

• Total time = 120 s

$:\implies\:\:\bf{Velocity_{(avg.)}\:=\:\dfrac{60000}{120}\:}$

$:\implies\:\:\bf\green{Velocity_{(avg.)}\:=\:500\:m/s}$

$\\ \Large\bf\orange{Therefore,}$

⑴ The average velocity is 500 m/s.