# Question Video: Finding the Kinetic Energy of a Body Mathematics

A body of mass 1 kg is moving in a straight line. After time 𝑡 seconds, where 𝑡 ≥ 0, the body’s displacement relative to a fixed point is given by vector 𝐬 = [(−6𝑡² − 2𝑡)𝐞] m, where 𝐞 is a fixed unit vector. Find the kinetic energy of the body, 3 seconds after it started moving.

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### Video Transcript

A body of mass one kilogram is moving in a straight line. After time 𝑡 seconds, where 𝑡 is greater than or equal to zero, the body’s displacement relative to a fixed point is given by vector 𝐬, which is equal to negative six 𝑡 squared minus two 𝑡 multiplied by 𝐞 meters, where 𝐞 is a fixed unit vector. Find the kinetic energy of the body three seconds after it started moving.

We are told in the question that vector 𝐬, the body’s displacement, is equal to negative six 𝑡 squared minus two 𝑡 multiplied by the unit vector 𝐞. We know that we can calculate the velocity vector by differentiating the displacement vector with respect to time. Differentiating negative six 𝑡 squared gives us negative 12𝑡. And differentiating negative two 𝑡 gives us negative two. d𝐬 by d𝑡 is equal to negative 12𝑡 minus two multiplied by the unit vector 𝐞. As our displacement was given in meters, the velocity vector will be in meters per second.

We can now calculate the velocity after three seconds by substituting 𝑡 is equal to three. Vector 𝐯 is equal to negative 12 multiplied by three minus two multiplied by the unit vector 𝐞 meters per second. Negative 12 multiplied by three minus two is equal to negative 38. The velocity vector after three seconds is equal to negative 38𝐞 meters per second.

As we want a scalar quantity for the rest of this question, the magnitude of the velocity or the speed is equal to 38 meters per second. We can now use this value to calculate the kinetic energy of the body, where the kinetic energy is equal to a half 𝑚𝑣 squared. 𝑚 is the mass in kilograms and 𝑣 the velocity or speed in meters per second. This will give us an answer for the kinetic energy in joules.

Substituting in the values for the mass and speed, we have a half multiplied by one multiplied by 38 squared. This is equal to 722. The kinetic energy of the body three seconds after it started moving is 722 joules.