# A rocket sled begins from rest. In the first 3 s, it goes 100 m at a constant acceleration. \\ a)...

## Question:

A rocket sled begins from rest. In the first 3 s, it goes 100 m at a constant acceleration.

a) What is the velocity at the end of the first 3 s?

b) What was the acceleration?

c) If the maximum velocity of the rocket sled is 290 {eq}\frac{m}{s}{/eq}, how long does it take to reach that velocity?

d) How far does the rocket sled go in that amount of time?

## Kinematics Equation

The kinematics equation are the set of equations used for motion of a object with uniform/constant acceleration. Few such equations are, {eq}v_f = v_i +at \\ x = (\dfrac{v_f+v_i}{2})t {/eq}, where

*x*is the displacement*t*is the time interval- {eq}v_i {/eq} and {eq}v_f {/eq} are the initial and final velocity of the particles.

## Answer and Explanation: 1

**Given :**

The initial velocity of the rocket is, {eq}v_i = 0 {/eq}

The displacement of the rocket in *3 s* is, *x* = 100 m

The time interval of the motion is, *t* = 3 s

**Part (a)**

Let the final velocity of rocket be, {eq}v_f {/eq}

Applying the kinematics equation, we get, $$\begin{align*} x &= (\dfrac{v_f+v_i}{2})t \\ 100 &= (\dfrac{v_f+0}{2})3 \\ v_f &= 66.67 \ \frac{m}{s} \end{align*} $$

**Part (b)**

Let the acceleration of the rocket be, *a*

Applying the kinematics equation, we get, $$\begin{align*} v_f &= v_i +at \\ 66.67 &= 0 +a(3) \\ a &= 22.22 \ \frac{m}{s^2} \end{align*} $$

**Part (c)**

The maximum velocity of rocket is, {eq}v_{max} = 290 \ \frac{m}{s} {/eq}

Let the time taken to reach the maximum velocity be, {eq}t_1 {/eq}

Applying the kinematics equation, we get, $$\begin{align*} v_{max} &= v_i +at_1 \\ 290 &= 0 +(22.22)t_1 \\ t_1 &= 13.05 \ s \end{align*} $$

**Part (d)**

Let the displacement of rocket when it reaches it's maximum velocity be, {eq}x_1 {/eq}

Applying the kinematics equation, we get, $$\begin{align*} x_1 &= (\dfrac{v_{max}+v_i}{2})t_1 \\ &= (\dfrac{290+0}{2})(13.05) \\ &= 1892.25 \ m \end{align*} $$

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Chapter 7 / Lesson 3Kinematic quantities are calculated using the 'big 5' equations that explain a motion in constant acceleration, when in a straight line. Learn each of these variables and how they fit into example calculations provided.