Video: Calculating the Displacement of a Particle given the Time and the Velocity Expression

A car, starting from rest, began moving in a straight line from a fixed point. Its velocity after 𝑡 seconds is given by 𝑣 = (8𝑡² + 6𝑡) m/s, 𝑡 ≥ 0. Calculate the displacement of the car when 𝑡 = 9 seconds.

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Video Transcript

A car, starting from rest, began moving in a straight line from a fixed point. Its velocity after 𝑡 seconds is given by 𝑣 equals eight 𝑡 squared plus six 𝑡 meters per second, where 𝑡 is greater than or equal to zero. Calculate the displacement of the car when 𝑡 equals nine seconds.

In order to work out an expression for the displacement, we need to integrate the expression for the velocity. In this case, the velocity was equal to eight 𝑡 squared plus six 𝑡 this means the displacement 𝑠 will be closer to the integral of eight 𝑡 squared plus six 𝑡. Our limits are zero and nine as the car started from rest and we want to calculate the displacement when 𝑡 equals nine seconds.

Integrating eight 𝑡 squared gives us eight 𝑡 cubed divided by three and integrating six 𝑡 gives us six 𝑡 squared divided by two, which can be simplified to three 𝑡 squared. Substituting in our limits gives us an answer of 2187.

This means that the displacement of the car when 𝑡 equals nine seconds is 2187 meters.

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