Question Video: Finding the Kinetic Energy and the Velocity of a Tram Towed by an Inclined Rope Using the Work–Energy Principle Mathematics

Video Transcript

A tram of mass two metric tons was being towed by a rope inclined at an angle of 60 degrees to the track against a resistance of 20 kilogram-weight. Given that the tension in the rope was 121 kilogram-weight, use the work energy principle to find the kinetic energy of the tram 𝐸 and its speed 𝑣 after moving a distance of 16 meters. Take the acceleration due to gravity to be 𝑔 is equal to 9.8 meters per second squared.

Let’s begin by sketching a diagram to model the situation in this question. We are told that there is a tram of mass two metric tons. Since there are 1000 kilogram in one metric ton, the tram has a mass of 2000 kilograms. The tram is being towed by a rope at an angle of 60 degrees to the horizontal. And this rope has a tension of 121 kilogram-weight. There is a resistance acting in the opposite direction to the motion of 20 kilogram-weight. By letting the positive direction be the direction of motion, we can calculate the horizontal force 𝐅 on the tram.

Using our knowledge of right angle trigonometry, we see that the horizontal component of the tension force, labeled 𝑥, is equal to the tension force multiplied by the cos of 60 degrees. Resolving horizontally, we see that the sum of the forces 𝐅 is equal to 121 multiplied by cos of 60 minus 20. The cos of 60 degrees is one-half, so the right-hand side simplifies to 60.5 minus 20. The horizontal force 𝐅 acting on the tram is therefore equal to 40.5 kilogram-weight.

We know that the work done on a body is equal to the force acting on a body multiplied by the distance traveled in that direction. We are told that the tram moves a distance of 16 meters. Therefore, we need to multiply 40.5 by 16. This is equal to 648 kilogram-weight meters. The work energy principle tells us that the work done is equal to the change in kinetic energy. The kinetic energy of the tram 𝐸 is therefore equal to 648 kilogram-weight meters.

We are also asked to calculate the speed of the tram at this point. We know that the kinetic energy is equal to a half 𝑚𝑣 squared, where 𝑚 is the mass measured in kilograms and 𝑣 the velocity in meters per second. In order to use this equation, we need our kinetic energy to be given in joules. We can do this by multiplying 648 by the gravity 9.8 meters per second squared. The kinetic energy 648 kilogram-weight meters is equivalent to 6350.4 joules.

We can now set this equal to a half 𝑚𝑣 squared, where the mass is 2000 kilograms. One-half of 2000 is 1000. We can then divide both sides of our equation by 1000 such that 𝑣 squared is equal to 6.3504. Square rooting both sides of this equation and taking the positive value, we have 𝑣 is equal to 2.52. The speed of the tram after moving a distance of 16 meters is 2.52 meters per second.

The two answers to this question are 𝐸 is equal to 648 kilogram-weight meters and 𝑣 is equal to 2.52 meters per second.