Question

The equations of motion of a rocket are: $x = 2t,y = - 4t,z = 4t$, where the time $t$ is given in seconds and the coordinate of a moving in kilometres. At what distance will the rocket be from the starting point $O\left( {0,0,0} \right)$ in 10 seconds?
A. 60 km
B. 30 km
C. 45 km
D. None of these

Answer 1

Hint: We will use the given equation of motion to find the position of the rocket after 10 seconds by substituting the value $t = 10$. Then, find the distance using the distance formula.

Complete step-by-step answer:
The equation of the motion of rocket is given as $x = 2t,y = - 4t,z = 4t$
When $t = 0$, the rocket is at origin and we have to calculate the distance when $t = 10$
Substitute $t = 10$ in the equation of motion to find the corresponding position of rocket after 10 seconds.
$
  x = 2\left( {10} \right),y = - 4\left( {10} \right),z = 4\left( {10} \right) \\
   \Rightarrow x = 20,y = - 40,z = 40 \\
$
The position vector of the rocket is given by
\[\overrightarrow r = x\hat i + y\hat j + z\hat k\]
Therefore, the corresponding position vector of the rocket after 10 seconds is
\[\overrightarrow r = 20\hat i - 40\hat j + 40\hat k\]
Also, the distance between any vector $\overrightarrow r $ and point $\left( {a,b,c} \right)$ is given by $\left| r \right| = \sqrt {{{\left( {x - a} \right)}^2} + {{\left( {y - b} \right)}^2} + {{\left( {z - c} \right)}^2}} $
We have to find the distance between by \[\overrightarrow r = 20\hat i - 40\hat j + 40\hat k\] and $O\left( {0,0,0} \right)$.

$\sqrt {{{\left( {x - a} \right)}^2} + {{\left( {y - b} \right)}^2} + {{\left( {z - c} \right)}^2}} $
$
   \Rightarrow \left| r \right| = \sqrt {{{\left( {20 - 0} \right)}^2} + {{\left( { - 40 - 0} \right)}^2} + {{\left( {40 - 0} \right)}^2}} \\
   \Rightarrow \left| r \right| = \sqrt {{{\left( {20} \right)}^2} + {{\left( { - 40} \right)}^2} + {{\left( {40} \right)}^2}} \\
   \Rightarrow \left| r \right| = \sqrt {400 + 1600 + 1600} \\
   \Rightarrow \left| r \right| = \sqrt {3600} \\
 $
On simplifying the above expression, we get,
$ \Rightarrow \left| r \right| = 60km$
Hence, option A is correct.

Note: We had used the distance formula, to find the distance between two points. The starting point of the rocket is $O\left( {0,0,0} \right)$ which is the origin. Thus, we have calculated the distance from origin to the position of the rocket after 10 seconds.