Homework SourseThe position of a particle at time t is given by rtcos4tisin
The position of a particle at time t, is given by r(t)=cos(4t)i-sin(4t)j+3k.
A). Find the velocity, acceleration, and speed of the particle.
B). Show that the particle moves with constant speed.
C). Find the angle between the particle\'s position and acceleration vector at time t=0.
A). Find the velocity, acceleration, and speed of the particle.
B). Show that the particle moves with constant speed.
C). Find the angle between the particle\'s position and acceleration vector at time t=0.
Solution
Velocity v(t) = dr/dt = -4sin(4t)i - 4cos(4t)j
Acceleration a(t) = dv/dt = -16cos(4t)i + 16sin(4t)j
Speed = |v| (magnitude of v(t) ) = ((-4sin(4t))2 + (-4cos(4t))2)
= (16sin2(4t) + 16cos2(4t)) = (16(sin2(4t)+cos2(4t))) =16 = 4 ........ (sin2()+cos2() = 1)
Hence we see the speed is independent of time
Hence speed of particle is constant
At t = 0;
r(0) = i +3k; a(0) = -16i
Now angle between the two can be calculated by using the dot product formula i.e A.B = |A||B|cos()
Hence here:
r.a = -16 ; |r| = (1+9) = 10 ; |a| = 16
Hence cos () = -16/(16*10) = 1/10
Hence = cos-1(1/10) =71.565 degrees
