Question

Hello can someone please please help with these:

1) A particle travels along the line y= 2x+1 with uniform speed . Find the components
of its velocity parallel to the axes of x and y .
2) The velocities of two particles at time are 2ti + 2j and 4i + (3-2t) j respectively.
Find the instant when the particles are moving in perpendicular directions.

Answer 1

Of course, I can help with these physics problems:

1) To find the components of velocity parallel to the x and y axes, you can use the fact that velocity is the rate of change of position. Given the equation of motion y = 2x + 1, you can differentiate it to find the velocity vector:

dy/dt = d/dt (2x + 1)

The velocity vector is represented as (dx/dt)i + (dy/dt)j. In this case:

(dx/dt)i = (2i) because the derivative of 2x with respect to x is 2.

Now, find dy/dt by differentiating 2x + 1 with respect to t:

dy/dt = d/dt (2x + 1) = 2(dx/dt)

So, dy/dt = 2(2i) = 4i.

The components of velocity parallel to the x and y axes are:

Vx = 2i

Vy = 4i

2) To find the instant when the particles are moving in perpendicular directions, their relative velocities should be perpendicular. The dot product of their velocity vectors should be zero.

Given the velocities:

Particle 1: V1 = 2ti + 2j

Particle 2: V2 = 4i + (3-2t)j

Calculate their relative velocity, V_rel = V1 - V2:

V_rel = (2ti + 2j) - (4i + (3-2t)j)

Now, take the dot product of V_rel with itself and set it equal to zero:

V_rel · V_rel = 0

(2ti + 2j - 4i - (3-2t)j) · (2ti + 2j - 4i - (3-2t)j) = 0

Expand and simplify this expression, and you'll find the value of t when the particles are moving in perpendicular directions.