Problem 2/61 Page 46, Engineering Mechanics - Dynamics, Meriam and Kraige, 4th Edition

Given: x = A t² - B t + C and y = D sin w t.

Find: The magnitude of the velocity at time t0, the magnitude of the acceleration at time t0, and the angle between the velocity and acceleration at time t0.

Relevant Parameters:

t0 = 3 s
A = 1 in/s²
B = 4 in/s
C = 20 in
D = 3 in
w = 2 rad/s

v = dr/dt = dx/dt i + dy/dt j

x = A t² - B t + C
dx/dt =2 A t - B

y = D sin w t.
dy/dt = w D cos w t

v = ( 2 A t - B ) i + ( w D cos w t ) j
v(t0) = ( 2 A t0 - B ) i + ( w D cos w t0 ) j = 2 in/s i + 5.76 in/s j
| v(t0) | = { 2² + 5.76² }^1/2 in/s  = 6.10 in/s

a = dv/dt = dvx/dt i + dvy/dt j

vx = 2 A t - B
dvx/dt = 2 A

vy =  w D cos w t
dvy/dt = - w² D sin w t
 
a = 2 A i - w² D sin w t j

a(t0) = 2 A i - w² D sin w t0 j = 2 in/s² i + 3.35 in/s² j

| a(t0) | = { 2² + 3.35² }^1/2 in/s² = 3.90 in/s²
 
 To evaluate the angle between the two vectors we can use the dot product.

a . v = | a | | v | cos(q) = ax vx + ay vy

cos(q) = { 2 * 2 + 3.35 * 5.76 } / { 3.90 * 6.10 } = 0.979

q = 11.67 degrees