To solve the problem of the stopping distances of the two trolleys, we need to apply some fundamental principles of physics, particularly Newton's laws of motion and the concept of friction. Let's break it down step by step.
Understanding the Scenario
We have two identical trolleys on a rough surface. One trolley (let's call it Trolley A) is being pushed by a man, and the other trolley (Trolley B) is stationary. The mass of Trolley A, including the man, is three times the mass of Trolley B. This means if we denote the mass of Trolley B as m, then the mass of Trolley A is 3m.
Identifying Forces and Friction
When Trolley A is pushed, it accelerates due to the force exerted by the man. However, both trolleys will eventually come to a stop due to the frictional force acting against their motion. The frictional force can be calculated using the formula:
- Frictional Force (f) = μ * N
Where μ is the coefficient of friction and N is the normal force. For horizontal surfaces, the normal force is equal to the weight of the object, which is mass times gravitational acceleration (g).
Calculating Stopping Distances
The stopping distance (d) of an object can be derived from the work-energy principle, which states that the work done by the frictional force is equal to the change in kinetic energy of the object. The formula for stopping distance is:
Where v is the initial velocity and a is the deceleration due to friction. The deceleration can be expressed as:
Applying the Ratios
Since Trolley A has a mass of 3m and Trolley B has a mass of m, we can analyze their stopping distances. The stopping distance for Trolley A (d₁) and Trolley B (d₂) can be expressed as:
- d₁ = (v₁²) / (2 * μ * g)
- d₂ = (v₂²) / (2 * μ * g)
Here, v₁ and v₂ are the initial velocities of Trolley A and Trolley B, respectively. Since Trolley B is initially at rest, its initial velocity is 0, and it will not move unless acted upon by a force. Therefore, we only need to consider Trolley A's motion.
Determining the Ratios
Now, if we assume that Trolley A pushes Trolley B and they both start moving together, the initial velocity of Trolley A will be greater than that of Trolley B. However, since Trolley A has a greater mass, it will experience a greater stopping distance due to the larger force of friction acting on it. The ratio of the stopping distances can be determined by comparing their masses:
- Ratio of stopping distances (d₁ : d₂) = (mass of Trolley A) : (mass of Trolley B)
Substituting the values, we get:
Final Thoughts
Thus, the ratio of the stopping distances of Trolley A to Trolley B is 3:1. This means that Trolley A, being heavier, will travel three times the distance before coming to a stop compared to Trolley B, assuming they both experience the same frictional force. This example illustrates how mass and friction play crucial roles in motion and stopping distances.
