The coordinates of a moving particle at time t are given by x=ct2 and y=bt2. The speed of the particle is given by
Question
The coordinates of a moving particle at time t are given by:
- x = ct²
- y = bt²
The speed of the particle is given by:
Solution
The speed of a particle is given by the magnitude of its velocity vector. The velocity vector of the particle is given by the derivative of its position vector with respect to time.
The x-component of the velocity is given by the derivative of x with respect to time, which is:
dx/dt = 2ct
Similarly, the y-component of the velocity is given by the derivative of y with respect to time, which is:
dy/dt = 2bt
The magnitude of the velocity vector, which is the speed of the particle, is given by the square root of the sum of the squares of the x and y components of the velocity. Therefore, the speed of the particle is:
v = sqrt((dx/dt)^2 + (dy/dt)^2) = sqrt((2ct)^2 + (2bt)^2) = sqrt(4c^2t^2 + 4b^2t^2) = sqrt(4t^2(c^2 + b^2)) = 2t*sqrt(c^2 + b^2)
So, the speed of the particle at time t is 2t*sqrt(c^2 + b^2).
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