Question

The x- coordinate of a particle depends on Time (E) as X = a sin (2t) + t cos(2t) then its speed V depends on x given by the equation​

Answer 1

The x - coordinate of a particle depends on time (t) as x = a sin(2t) + a cos(2t)

To find : speed of the particle depends on x given by the equation....

solution : equation of position of particle along x - axis is given by, x = a sin(2t) + a cos(2t) ......(1)

differentiating x with respect to time we get,

dx/dt = d(a sin(2t) + a cos(2t))/dt

⇒ v = 2a cos(2t) - 2a sin(2t)

⇒v = 2[a cos(2t) - a sin(2t)]

⇒v/2 = [a cos(2t) - a sin(2t)] ......(2)

now squaring equations (1) and (2) and then adding we get,

x² + v²/4 = {a sin(2t) + a cos(2t)}² + [{a cos(2t) - a sin(2t)}]²

⇒x² + v²/4 = 2a²[sin²(2t) + cos²(2t)] + 2a²sin(2t) cos(2t) - 2a²sin(2t) cos(2t)

⇒x² + v²/⁴ = 2a² × 1 + 0

⇒x² + v²/4 = 2a²

⇒v²/4 = 2a² - x²

⇒v² = 8a² - 2x²

⇒v = √(8a² - 2x²)

Therefore the speed of the particle depends on x is √(8a² - 2x²)