Question

The Equation Rt) (3t+4)+ μf(i)+ (t2-9)+(4t) K Is The Position Of A Particle In Space At Time T Find The Partice's Velocity And Acceleration Vectors.Then write the particle's velocity at t=1 as a product of it's speed and direction.

Answer 1

The velocity vector and acceleration vector of the particle can be found by differentiating the position function with respect to time. At t = 1, the velocity of the particle can be expressed as the product of its speed and direction.

Step-by-step explanation:

The position of a particle in space at time t is given by the equation r(t) = (3t+4)î + μf(i) + (t^2-9)ĵ + (4t)K. To find the particle's velocity and acceleration vectors, we need to differentiate the position function with respect to time. Differentiating each term separately:

Velocity vector, v(t) = (3î + 0 + 2tĵ + 4K)m/s

Acceleration vector, a(t) = (0î + 0ĵ + 2ĵ + 4K)m/s^2

At t = 1, the velocity is v(1) = (3î + 0 + 2(1)ĵ + 4K) = 3î + 2ĵ + 4K. The speed of the particle is the magnitude of the velocity vector, so the speed at t = 1 is |v(1)| = sqrt(3^2 + 2^2 + 4^2) = sqrt(29). The direction of the velocity vector is the unit vector in the direction of the velocity, so the direction at t = 1 is v(1)/|v(1)| = (3/√29)î + (2/√29)ĵ + (4/√29)K.