A car which has run out of petrol is being towed by a breakdown truck along a straight horizontal road - Edexcel - A-Level Maths Mechanics - Question 8 - 2003 - Paper 1

To find the tension (T) in the rope, we consider the forces acting on the car alone:

1. The driving force provided by the truck is T.

2. The resistive force acting on the car is 400 N.

3. Applying Newton's second law for the car:

   $T - 400 N = 800 kg * a$

   Substitute the acceleration from part (a):

   $T - 400 N = 800 kg * 0.7 m s^{-2}$

   Therefore:

   $T = 400 N + 560 N = 960 N$

After the rope breaks, we analyze the forces acting on the truck:

1. The driving force remains 2400 N, while the resistive force is 600 N.

2. The net force acting on the truck is:

   $F_{net} = 2400 N - 600 N = 1800 N$

3. The acceleration a' of the truck can be found:

   $a' = \frac{F_{net}}{m_{truck}} = \frac{1800 N}{1200 kg} = 1.5 m s^{-2}$

4. To find the time taken to reach 28 m s⁻¹: Using the formula: $v = u + a't$ (where u = 20 m/s, v = 28 m/s):

   $28 = 20 + 1.5t \Rightarrow t = \frac{8}{1.5} = 5.33 s$

5. If the rope had not broken, the time taken can be calculated similarly:

   With acceleration of 0.7 m/s²:

   $v = u + at \Rightarrow 28 = 20 + 0.7t \Rightarrow t = \frac{8}{0.7} = 11.43 s$

6. Finally, the difference in time:

   $11.43 s - 5.33 s \approx 6.1 s$ Therefore, the truck reaches the desired speed approximately 6 s earlier.