Aparticle follows a planar path defined by x = kξ, y = 2k[1 − exp(ξ )], such that its speed is v = βξ, where k and β are constants. determine the velocity and acceleration at ξ = 0.5

The velocity of the particle is given by the derivative of the position vector . We have

with derivative

At , we get a velocity vector of

Not important as far as I can tell, but the particle's speed at is , which satisfies

(Perhaps you may be required to solve for in terms of , then report the velocity/acceleration vectors with respect to , but I'll not do so because the original question makes no mention of needing to do that.)

Acceleration is given by the derivative of the velocity vector:

and at , the acceleration vector is