phy (circular motion+momentum)

1. In order for a car to travel in circular motion in a road, a radial acceleration is required to provide the centripetal force needed. In fact, such acceleration changes the direction of the car so that it can travel in circular motion. The required acceleration is v^2/r, where v is the linear velocity and r is the radius of curvature, with direction perpendicular to v. If other forces are insufficient to make the required acceleration, then the friction provides the rest of the centripetal force needed. Therefore, the friction is really helping the car to move in a circle, but not opposing the motion.

2. A bird have to roll its body so that a horizontal component perpendicular to its velocity can be obtained from the lifting force. This component contributes to the force required for radial acceleration. The magnitude of this component will determine the radius of curvature.

3. Angular displacement is defined as angular speed multiplied by time, and the unit is radians. There are only two possible directions, positive and negative, like a scalar. Since one complete revolution rotates along the same direction on the way, it should be regarded as having moved forward 2π radians. Zero angular displacement would imply a rotational motion that started forward but later go backward with the same change in angle. An example is a D.C. motor without a commutator. It would rotate in one direction and then reverse the rotation shortly.

In fact, the angle displacement of the spacecraft is 2π*14 = 88.0 rad

4. Let u be the final common velocity
By conservation of momentum,
3vm + 2vm = u(m+m) (Move together therefore same speed)
5vm = 2um
u = 2.5v

5. Let the mass of Ball A be M and that of Ball B be m
Let the final velocity of A be v1 and that of B be v2
Then Mv + 0m = Mv1 + mv2
and 0.5Mv^2 = 0.5Mv1^2 + 0.5mv2^2

Solving, v1 = v(M-m)/(M+m), v2 = 2Mv/(M+m)
Since M >> m, m/M ≈ 0
v1 = v(1-m/M)/(1+m/M) ≈ v
v2 = 2v/(1+m/M) ≈ 2v

2013-06-03 23:06:51 補充：
For Q1, only the component of friction tangential to the car's circular motion is opposing the motion of the car. Any component of friction perpendicular to the direction of the velocity will not oppose the motion since they do not do work on the car and take away the kinetic energy from it.

2013-06-03 23:10:38 補充：
In fact, the car always has the tendency to move with constant velocity due to inertia. Unless forces perpendicular to the car's velocity act on it, the car will not move in a circular path. Therefore it is better to say the car have no tendency to go around in circles automatically.