AMath 1B Additional Problems 3 week 4-2025 t is in seconds.
Determine the magnitudes of the boat’s
velocity and acceleration when it has traveled 20 m.
1. A car is driving around a curve of radius 200 m, while 7. The driver of the car maintains a constant speed of
increasing its speed at the rate of 0.8 m/s2 . At a cer- 40 m/s. Determine the angular velocity of the camera
tain instant, the magnitude of the total acceleration is tracking the car when θ = 15◦ .
measured to be 1.5 m/s2 . What is the speed of the car
at that instant measured in km/h?
2. A particle travels along a plane curve from a point A to
a point B. The path length between A and B is 2 m.
The speed of the particle is 4 m/s at A and 2 m/s at B.
The rate of change of the speed is constant. (a) Find
the tangential component of the acceleration when the
particle is at B (b) If the magnitude of the acceleration 8. As it passes the indicated position the particle P has
at B is 5 m/s2 , determine the radius of curvature of the constant speed v = 100 m/s along the straight line
path at B shown. Determine the corresponding values for ṙ, θ̇, r̈
3. The car travels at a constant speed from the bottom and θ̈. (Ans: ṙ = −96.6 m/s, θ̇ = 0.229 rad/s, r̈ = 5.92
A of the dip to the top B of the hump. If the radius m/s2 and θ̈ = 0.39rad/s2 )
of curvature of the road at A is ρA = 120 m and the
car acceleration at A is 0.4g, determine the car speed
v. If the acceleration at B must be limited to 0.25g,
determine the minimum radius of curvature ρB of the
road at B. (Ans: v = 21.6 m/s, ρB = 190.4 m)
9. The smooth bar rotates in a horizontal plane at a con-
stant angular velocity of ω0 = 8 rad/s. The radial com-
ponent of the acceleration of collar C is given by ar −6r
m/s2 . When r = 1.5 m, the radial component of the
4. The particle passes point O at a speed of 8 m/s. Be- velocity of C is vr = 3 m/s. Determine the velocity and
√ acceleration of C in polar coordinates when r = 2 m.
tween O and B the speed increases at a rate of 4 v
m/s2 , where v is in m/s. Determine the magnitude of (Ans: v = 10.51er +16eθ m/s and a = −12er +168.16eθ
the acceleration when the particle is (a) just to the left m/s2 )
of point A, (Ans: 13.75 m/s2 ) (b) just to the right of
point A. (Ans: 22.24 m/s2 )
10. The racing airplane is beginning an inside loop in the
vertical plane. The tracking station at O records the
5. At the instant shown the magnitude of the airplane’s following data for the particular instant: r = 90 m,
velocity is 130 m/s, its tangential component of accel- ṙ = 15.5 m/s r̈ = 74.5 m/s2 , θ = 30◦ , θ̇ = 0.53 rad/s
¨ = −0.29 rad/s2 . Obtain the values of v, v̇, ρ
and ]theta
eration is at = −4 m/s2 and the rate of change of its
path angle is 5◦ /s. (a) What are the airplane’s velocity and β at this instant. (Ans: v = 50.2m/s, v̇ = 6.01
and acceleration in terms of normal and tangential com- m/s2 , ρ = 50.5 m and β = 12◦ )
ponents? (Ans: v = 130et m/s, a = −4et + 11.34en
m/s2 ) (b) What is the instantaneous radius of curva-
ture of the airplane’s path at this instant ? (Ans: 1490
m)
6. Starting from rest the motorboat travels around the
circular path, ρ = 50 m, at a speed v = 0.8t m/s, where
