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Motion MATH in space: Velocity and Acceleration
SECTION 14.4
Velocity and Acceleration: Suppose that a particle moves through space so that its position vector at any time t is given by r(t). Then the velocity vector v(t) at time t is given by v(t) = lim =
h0
r(t + h) r(t) . h
The velocity vector is also the tangent vector and points in the direction of the tangent line. The magnitude of the velocity vector, v(t) , is the speed of the particle at time t. The acceleration os the particle is dened as the derivative of the velocity. Therefore, a(t) = v (t) = r (t). From the vector integrals in section 14.2, we have r(t) = v(t) dt and v(t) = a(t) dt.
example 1: Find the velocity, acceleration, and speed of a particle with position vector r(t) = 2 5, 4 t . Sketch the path of the particle and draw the velocity and acceleration vectors for t = 1.
example 2: Find the acceleration, velocity, and speed of a particle with position function r(t) = t sin ti + t cos tj + t2 k.
example 3: Find the velocity and position vectors of a particle that has acceleration vector a(t) = ti + t2 j + cos 2tk and initial velocity v(0) = i + k and initial position r(0) = 2i + 3j.
Tangential and Normal Components of Acceleration: We will now resolve the acceleration into two components one in the direction of the tangent and one in the direction of the normal. If we let v = v(t) , the tangential component of acceleration, denoted aT , is given by aT = v = va r (t) r (t) = v r (t)
The normal component of acceleration, denoted aN , is given by aN = v 2 = r (t) r (t) r (t)
example 4: A particle moves with position function r(t) = ti + t2 j + 3tk. Find the tangential and normal components of acceleration.
Homework: pp 914915; 315 odd, 19, 22, 3135 odd