At t = √2 sec.
there velocity vectors perpendicular to each other

Problem:
Two particles are projected from a tower horizontally in opposite directions with velocities 20m/s and 10m/s. Find the time when their velocity vectors are mutually perpendicular. Take g=10m/s^2.

Solution:

To find the time when the velocity vectors of the two particles are mutually perpendicular, we need to analyze the motion of each particle separately and then determine the point in time where their velocities are perpendicular.

Particle 1:
The first particle is projected horizontally with a velocity of 20m/s. Since there is no vertical component to its initial velocity, the acceleration due to gravity will not affect its horizontal motion. Therefore, the horizontal velocity of the first particle remains constant throughout its motion.

Particle 2:
The second particle is projected horizontally in the opposite direction with a velocity of 10m/s. Similar to the first particle, the horizontal velocity of the second particle remains constant throughout its motion.

Analyzing the motion:
Let's assume that the particles are projected at time t=0, and we want to find the time (t) when their velocity vectors are mutually perpendicular. To do this, we need to find the position vectors of the particles at time t and then determine if their velocity vectors are perpendicular.

Let's consider the motion of the particles in the x-axis (horizontal direction) and y-axis (vertical direction).

Particle 1:
The position vector of the first particle at time t is given by:
x1(t) = 20t (since the horizontal velocity is constant)
y1(t) = 0 (since there is no vertical component)

Particle 2:
The position vector of the second particle at time t is given by:
x2(t) = -10t (since the horizontal velocity is constant in the opposite direction)
y2(t) = 0 (since there is no vertical component)

Now, let's find the velocity vectors of the particles at time t:

Particle 1:
The velocity vector of the first particle is given by:
v1(t) = (dx1(t)/dt) i + (dy1(t)/dt) j
= 20 i + 0 j
= 20i

Particle 2:
The velocity vector of the second particle is given by:
v2(t) = (dx2(t)/dt) i + (dy2(t)/dt) j
= -10 i + 0 j
= -10i

Determining if the velocity vectors are perpendicular:
To determine if the velocity vectors of the particles are perpendicular, we need to calculate their dot product and check if it is zero.

v1(t) · v2(t) = (20i) · (-10i)
= -200

Since the dot product is not zero, the velocity vectors are not perpendicular.

Conclusion:
Based on the analysis, the velocity vectors of the particles never become mutually perpendicular. Therefore, there is no time when their velocity vectors are perpendicular.