# Video: Finding the Distance Covered by a Cyclist Where They Moved with Uniform Acceleration Then at Uniform Velocity

A cyclist, riding down a hill, was accelerating at a rate of 0.5 m/s². By the time he reached the bottom of the hill, he was travelling at 1.5 m/s. He continued travelling at this speed for another 9.5 seconds. Determine the total distance 𝑠 that the cyclist covered.

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### Video Transcript

A cyclist, riding down a hill, was accelerating at a rate of 0.5 metres per second squared. By the time he reached the bottom of the hill, he was travelling at 1.5 metres per second. He continued travelling at this speed for another 9.5 seconds. Determine the total distance, 𝑠, that the cyclist covered.

To calculate the distance that the cyclist covered in the first part of the journey, we will use one of the equations of motion, or SUVAT equations. 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠, where 𝑣 is the final velocity, 𝑢 is the initial velocity, 𝑎 is the acceleration, and 𝑠 is the displacement.

We are told that the cyclist is accelerating at a rate of 0.5 metres per second squared and that his final velocity at the bottom of the hill is 1.5 metres per second. Therefore, 𝑢 is equal to zero. 𝑣 is equal to 1.5. And 𝑎 is equal to 0.5.

Substituting these values into the equation 𝑣 squared equals 𝑢 squared plus two 𝑎𝑠 gives us 1.5 squared is equal to zero squared plus two multiplied by 0.5 multiplied by 𝑠. 1.5 squared is equal to 2.25. And two multiplied by 0.5 is one. Therefore, 2.25 equals one 𝑠. This means that the cyclist covered 2.25 metres in the first part of his journey when he was riding down the hill.

For the second part of the journey, the cyclist is travelling at a constant speed. This means that we can use the equation distance equals speed multiplied by time, or velocity multiplied by time.

As his speed was 1.5 metres per second and the time was 9.5 seconds, we can multiply 1.5 by 9.5. This is equal to 14.25 metres. Therefore, the distance travelled in the second part of the journey is 14.25 metres.

To calculate the total distance, we need to add 2.25 and 14.25. This gives us an answer of 16.5 metres. Therefore, the cyclist travelled a total distance of 16.5 metres.