Irodov Problem 1.102

Let the mass of the body be m and let the force of friction acting on it be F. F will be proportional to N and be equal to Nax. The forces acting on the body are shown in the Figure. We can resolve all the forces along parallel and perpendicular direction to the slope.


Along the Perpendicular Direction there are two forces, i) the Normal reaction from the surface N and ii) the component of gravity and so we have,



Along the Parallel Direction there are two forces acting, i) the component of gravity and ii) the force of friction Nkx. Let the acceleration of the mass be w. So we have,



From (1) and (2) we have,











When the mass finally comes to stop v=0, putting this in (3) we have .

The maximum velocity occurs when acceleration has become 0 (w=0) and is about to reverse its direction. This occurs when . Substituting the value in (3) we have the maximum velocity as .

Irodov Problem 1.101

Since the resistance to the bullet is proportional to the squre of its velocity, it deceleration must be proportional to the square of its velocity. In other words,



Here, K is a positive proportionality constant. Let x represent the distance travelled by the bullet in the plank. From (1) we have,










To calculate time the bullet took to travel through the plank we take the following approach. From (1) we have,

Irodov Problem 1.100

a) From (1) it is clear that the boat will never completely stop, it will stop only after infinite time.
b) From (2) by setting v=0 (the time when the boat stops) we can find the distance covered as,
c) the time average of velocity (mean velocity) can be evaluated by . If the the velocity decreases time then from (1) we have


Now we can calculate the mean velocity as,

Irodov Problem 1.99

From (1) it is clear that the particle will come to stop (v=0) for the first time when . The distance covered during this time can be computed using (2) as





Also from (1) the maximum velocity will be .

Irodov Problem 1.98

Basically distance traveled s is different from displacement as its not the final position but the absolute length traversed and is given by the integral of modulas of velocity (|v|) rather than v.

Irodov Problem 1.97

a) Force is rate of change of momentum and so we have,






The force will cease act when F=0 or and from (1) thus we have .

b) P = mv and and so we have,









When the force stops at we have the distance traveled from (2) as .

Irodov Problem 1.96

a) Force is rate of change of momentum and here the force if constant equal to mg. In other words the particle's momentum changes at a constant rate of mg. Clearly in an interval of t seconds the momentum change will be mgt.

b)The component of velocity of the particle along the direction of gravity will be . Here, g is the vector of gravitational acceleration, g is its magnitude, vo is the velocity vector and vo is its magnitude. It can be found in any standard physics book (in projectile motion) that the total time before the particle hits the ground is given by . As in the part a), the rate of change in momentum is constant and equal to mg.
Thus the total change in momentum is given by .