Motion of Centre of Mass

A bullet of mass m moving at a speed v hits a ball of mass M kept at rest. A small part having mass m breaks from the ball and sticks to the bullet. The remaining ball is found to move at a speed v₁ in the direction of the bullet. Find the velocity of the bullet after the collision. 

Read each statement below carefully, and state, with reasons, if it is true or false;

The instantaneous acceleration of the point of contact during rolling is zero.

A ball falls on an inclined plane of inclination θ from a height h above the point of impact and makes a perfectly elastic collision. Where will it hit the plane again? 

To accelerate a car we ignite petrol in the engine of the car. Since only an external force can accelerate the centre of mass, is it proper to say that "the force generated by the engine accelerates the car"?

A fat person is standing on a light plank floating on a calm lake. The person walks from one end to the other on the plank. His friend sitting on the shore watches him and finds that the person hardly moves any distance because the plank moves backward about the same distance as the person moves on the plank. Explain.

Solve the previous problem if the coefficient of restitution is e. \[\text{ Use } \theta = 45^\circ\, e = \frac{3}{4} \text{ and h = 5 m } .\]

When a fat person tries to touch his toes, keeping the legs straight, he generally falls. Explain with reference to the following figure.