Question

A rope is attached from a truck to a \(1400-\mathrm{kg}\) car. The rope will break if the tension is greater than \(2500 \mathrm{N}\) Ignoring friction, what is the maximum possible acceleration of the truck if the rope does not break? Should the driver of the truck be concerned that the rope might break?

Short answer

The maximum acceleration of the truck without breaking the rope is 1.7857 m/s². The driver should be cautious not to accelerate too rapidly to prevent the rope from breaking, but with careful driving, the rope should not break.

Step-by-step solution

Write the equation for Newton's second law of motion for the car

Since we're interested in the car's maximum acceleration without breaking the rope, let's focus on the car for now and apply Newton's second law to it: \(F = m \times a\) Here, F is the tension force in the rope, m is the mass of the car, and a is the acceleration of the car.

Set the tension equal to the maximum allowed value

The rope can withstand a maximum tension force of 2500 N. Therefore, let's set F equal to this value: \(2500 = 1400 \times a\)

Solve for the acceleration of the car

Now, we need to solve for the acceleration, a: \(a = \frac{2500}{1400} = 1.7857 \frac{\mathrm{m}}{\mathrm{s^2}}\) So, the maximum acceleration of the car without breaking the rope is 1.7857 m/s².

Analyze whether the truck driver should be concerned

Now that we know the maximum acceleration of the car, we can compare it with the typical acceleration values for vehicles. A typical car accelerates from 0 to 60 mph (0 to 26.8224 m/s) in about 10 seconds, with an average acceleration of 2.68224 m/s². The maximum acceleration of the 1400 kg car without breaking the rope is 1.7857 m/s², which is lower than the average acceleration value mentioned above. However, keep in mind that the truck driver controls the acceleration – as long as the driver doesn't accelerate too rapidly, the rope should not break. To sum up, the maximum acceleration of the truck without breaking the rope is 1.7857 m/s². The driver should be cautious not to accelerate too rapidly to prevent the rope from breaking, but with careful driving, the rope should not break.