Question Video: Finding the Magnitude of the Resistance on a Body on a Rough Horizontal Plane Moving with Uniform Acceleration under a Horizontal Force Mathematics

Video Transcript

A car of mass 3.6 metric tons was moving along a straight horizontal road at 18 kilometers per hour when its engine cut out. Given that it traveled a further 180 meters before it stopped moving, find the magnitude of the resistance to the movement of the car.

We will begin by sketching a diagram of the scenario in this question. We are told that the mass of the car is 3.6 metric tons. Recalling that one ton is equal to 1000 kilograms, we can convert this to kilograms by multiplying by 1000. The mass of the car is therefore equal to 3600 kilograms. We are told that the car’s engine cuts out, which means there will be no driving or forward force. And we have been asked to calculate the magnitude of the resistance to the movement of the car.

Recalling Newton’s second law 𝐹 equals 𝑚𝑎, we know that the sum of our forces is equal to the mass multiplied by the acceleration. At present, we do not know the acceleration of the car. However, we can calculate this from the other information given. We will do this using our equations of motion or SUVAT equations. The speed of the car when the engine cut out was 18 kilometers per hour. So we will let the initial velocity 𝑢 be 18 kilometers per hour. The car traveled a further 180 meters. So this is the displacement 𝑠. As the car came to rest, the final velocity 𝑣 is zero kilometers per hour.

We are trying to calculate the value of 𝑎, the acceleration, in meters per second squared. Before using one of our equations, we need to convert the velocities from kilometers per hour to meters per second. We know that there are 1000 meters in one kilometer, and there are 3600 seconds in one hour. We can therefore convert from kilometers per hour to meters per second by multiplying by 1000 and then dividing by 3600. This is the same as dividing by 3.6.

18 divided by 3.6 is five. And zero divided by 3.6 is zero. The initial velocity is five meters per second, and the final velocity, zero meters per second. We will use the equation 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠. Substituting in our values, we have zero squared is equal to five squared plus two multiplied by 𝑎 multiplied by 180. This simplifies to zero is equal to 25 plus 360𝑎. We can then subtract 25 from both sides and divide through by 360 such that 𝑎 is equal to negative 25 over 360. This simplifies to negative five over 72.

We now have the acceleration and mass of the car. Since the friction or resistance force 𝐅 𝑟 is acting against the motion, we have negative 𝐅 𝑟 is equal to 3600 multiplied by negative five over 72. This means that negative 𝐅 𝑟 is equal to negative 250, and 𝐅 𝑟 is equal to 250. The magnitude of the resistance to the movement of the car is 250 newtons.