Introduction
This problem deals with the effect of friction on a moving object. When a box moves across a rough surface, the frictional force does negative work, reducing the speed of the box. We’ll apply the work-energy theorem to solve the two parts of this problem.
Given Data
- Mass of box, m = 8.0 kg
- Initial speed, u = 10 m/s
- Frictional force, f = 70 N
- Length of rough surface, s = 3.0 m
- We are asked to find:
- (a) Final speed of the box after covering 3.0 m
- (b) Distance required to bring the box to rest
Step 1: Work-Energy Theorem
Work done by friction = Change in kinetic energy
Work done = -f × s (negative because friction opposes motion)
Initial K.E. = (1/2)mu² = 0.5 × 8 × (10)² = 400 J
Work done = -70 × 3 = -210 J
Change in K.E. = Final K.E. – Initial K.E. = Work done
Final K.E. = 400 – 210 = 190 J
Final speed, v is given by:
½mv² = 190 → v² = (2 × 190)/8 = 47.5 → v = √47.5 ≈ 6.89 m/s
Step 2: Distance to Bring the Box to Rest
We now want the box to stop completely (final velocity = 0).
Initial K.E. = ½ × 8 × 10² = 400 J
To bring the box to rest, all this energy must be removed by friction:
Work done by friction = -f × s = -70 × s
So, -70 × s = -400 → s = 400 / 70 ≈ 5.71 m
Final Answers
- Speed of box after 3 m: 6.89 m/s
- Distance required to stop box: 5.71 m
Conclusion
This problem demonstrates how friction reduces the motion of an object. By using energy concepts, we determined that the speed of the box reduces to approximately 6.89 m/s after 3 m, and it would take around 5.71 m of rough surface to bring it to a complete stop.