Final answer:
The particle's velocity function v(t) is (50 m/s)î - (9.8 m/s²)t ĸ and the acceleration function a(t) is (0)î - (9.8 m/s²) ĸ. The initial conditions are r(0) = 0î + 0ĸ and v(0) = (50 m/s)î.
Step-by-step explanation:
The position of a particle is given by r(t) = (50 m/s)tî – (4.9 m/s²)t² ĸ. When we want to find the particle's velocity and acceleration as functions of time, we take the first and second derivatives of the position function with respect to time, respectively.
Velocity as a function of time, v(t), is the first derivative of r(t):
v(t) = dr(t)/dt = 50 m/s î - 2*(4.9 m/s²)t ĸ = (50 m/s)î - (9.8 m/s²)t ĸ.
Acceleration as a function of time, a(t), is the derivative of velocity which is the second derivative of r(t):
a(t) = dv(t)/dt = d/dt[(50 m/s)î - (9.8 m/s²)t ĸ] = (0)î - (9.8 m/s²) ĸ.
The initial conditions to produce this motion are found by evaluating the position and velocity functions at t = 0:
Position at t=0: r(0) = (0)î + (0)ĸ = 0î + 0ĸ.
Velocity at t=0: v(0) = (50 m/s)î - 0ĸ = (50 m/s)î.