A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram - AQA - A-Level Maths Mechanics - Question 17 - 2018 - Paper 2

To find the tension T in the rope, we again apply Newton's second law on the roller-skater.

1. The net force acting on the roller-skater is:

   $F_{net} = T - R$

   Where:

   • Mass of the roller-skater = 72 kg.
   • Acceleration = 0.2 m/s².

   Thus, we can write:

   $F_{net} = 72 imes 0.2 = 14.4 N$

   Setting the forces gives:

   $14.4 N = T - 78 N$

   Rearranging the equation yields:

   $T = 14.4 N + 78 N$

   Therefore:

   $T = 92.4 N$

A necessary assumption made in this problem is that the rope connecting the buggy and the roller-skater is massless, and that friction or air resistance does not significantly affect the system.

To determine whether the roller-skater stops before reaching the stationary buggy after releasing the rope at speed of 6 m/s, we will first calculate the deceleration experienced due to the resistance force.

1. We can write the equation of motion as:

   $T - R = ma$

   Where:

   • R = 78 N.
   • T = 0 (no tension after releasing).
   • Mass of the roller-skater = 72 kg.
   • Hence, we apply Newton's law to find acceleration:

   $0 - 78 N = 72 imes a$

   Solving for a gives:

   a = -rac{78}{72} = -1.083 ms^{-2}

2. To find the stopping distance (s), we can use the kinematic equation:

   $v^2 = u^2 + 2as$

   Where:

   • Final velocity v = 0 (when the roller-skater stops).
   • Initial velocity u = 6 m/s.
   • Using the calculated a:

   $0 = (6)^2 + 2(-1.083)s$

   Rearranging gives:

   s = rac{36}{2 imes 1.083} \\ s = 16.63 m

Since the stopping distance (16.63 m) is less than the distance to the buggy (20 m), the roller-skater will indeed stop before reaching the stationary buggy.

Upon releasing the rope, the roller-skater continues to experience resistance while moving forward. As the roller-skater loses the tension that was aiding their motion, a combination of their initial momentum and the resisting forces results in a slowing motion. The driver notices that the buggy begins to change speed, as it continues to travel with an initially applied driving force, leading to the buggy eventually coming to a stop at a distance of 20 m from the roller-skater's release point.