A particle moving in xy plane whose x and y coordinate vary with time, which is given as x=(1-cos 6t) y=sin 6t , find Distance travel by particle in 4 sec ?

Question

Answer

X=(1-cos6t)
Vx = dX /dt= 0-6×-sin6t = 6sin6t

Y= sin6t
Vy = dy/dt=6cos6t

v=√Vx^2+Vy^2 =√6sin6t^2 +√6cos6t^2
v=6√sin6t^2 √cos6t^2 =6×1 =6m/s

dist.= v×t=6×4=24m

Answer

At time 0 x=0 and y=0 .. then we find velocity  by differentiation.  Vx= 6 Sin6t  and Vy =6 Cos6t.. then V =√Vx^2 +Vy^2.. OK then we get the velocity is 6 m/s. then distance is 6×4 = 24m.. ok.

Answer

Given information:
The x-coordinate of the particle is given by x = (1 - cos 6t) and the y-coordinate is given by y = sin 6t.

To find:
The distance traveled by the particle in 4 seconds.

Approach:
To find the distance traveled by the particle, we need to calculate the total displacement of the particle in the given time interval and then calculate the magnitude of this displacement vector.

Calculating the displacement vector:
The displacement vector of the particle can be calculated by subtracting the initial position vector from the final position vector. Let's consider the initial time t = 0 and the final time t = 4.

Initial position vector: r₁ = (x₁, y₁) = ((1 - cos 6(0)), sin 6(0)) = (0, 0)
Final position vector: r₂ = (x₂, y₂) = ((1 - cos 6(4)), sin 6(4))

Calculating x and y coordinates at t = 4:
Substituting t = 4 in the given equations:
x₂ = (1 - cos 6(4))
y₂ = sin 6(4)

Calculating the displacement vector:
The displacement vector is given by:
Δr = r₂ - r₁ = (x₂ - x₁, y₂ - y₁)

Calculating the magnitude of the displacement vector:
The magnitude of the displacement vector is given by:
|Δr| = sqrt((x₂ - x₁)^2 + (y₂ - y₁)^2)

Calculating the distance traveled:
The distance traveled by the particle is equal to the magnitude of the displacement vector.

Final Solution:
The distance traveled by the particle in 4 seconds is |Δr|, where:
r₁ = (0, 0)
r₂ = ((1 - cos 6(4)), sin 6(4))
Δr = r₂ - r₁ = ((1 - cos 6(4)), sin 6(4)) - (0, 0)
|Δr| = sqrt((1 - cos 6(4))^2 + (sin 6(4))^2)

Simplifying the above expression will give you the final answer for the distance traveled by the particle in 4 seconds.

Answer

1/2=0.05left=8/8=1=left

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