Question

Two particles having position vectors \[{\vec r_1} = \left( {3\hat i + 5\hat j} \right)\] and \[{\vec r_2} = \left( { - 5\hat i - 3\hat j} \right)\] are moving with velocities \[{\vec v_1} = \left( {4\hat i + 3\hat j} \right)\] and \[{\vec v_2} = \left( {\alpha \hat i + 7\hat j} \right)\] If they collide after 2 seconds, the value of a is
A. \[2\]
B. \[4\]
C. \[6\]
D. \[ - 8\]

Answer 1

Hint: We can start by writing down the given data from the question. Then we can find the total distance travelled by adding the product of velocity and time with the position from which the body starts moving. The distance travelled by both the particles before collision will be the same and hence, we can equate them and find the value of the unknown thus solving the question.

Formulas used:
The distance travelled by a body at a given value of time and velocity is given by the formula,
\[S = vt\]
Where \[v\] is the velocity with which the body moves and \[t\] is the time taken for the motion.

Complete step by step answer:
Let us start by writing down the data given in the question. The particles are at a point,
\[{\vec r_1} = \left( {3\hat i + 5\hat j} \right)\] and \[{\vec r_2} = \left( { - 5\hat i - 3\hat j} \right)\]
The velocities of the particles are \[{\vec v_1} = \left( {4\hat i + 3\hat j} \right)\] and \[{\vec v_2} = \left( {\alpha \hat i + 7\hat j} \right)\]
The particles collide after a time \[t\] and travel equal distances.We can find the distance travelled in the time using the formula,
\[S = vt\]
\[\Rightarrow {\vec r_1} + {\vec v_1}t = {\vec r_2} + {\vec v_2}t\]
We can substitute the values and get
\[\left( {3\hat i + 5\hat j} \right) + \left( {4\hat i + 3\hat j} \right)t = \left( { - 5\hat i - 3\hat j} \right) + \left( {\alpha \hat i + 7\hat j} \right)t\]
Simplifying and bringing the like terms to one side we get,
\[16\hat i = 2\alpha \hat i\]
The unknown value can be found out as, \[\alpha = - 8\].

Therefore, the correct answer is option (D).

Note: The position vector, straight-line having one end fixed to a body and the other end attached to a moving point and used to describe the position of the point relative to the body. As the point moves, the position vector will change in length or in direction or in both length and direction. The length, direction, and orientation of the vector are the complete information that determines the translation.