Video: Finding the Expression for a Particle’s Displacement in Terms of Time given Its Acceleration Expression

A particle is moving in a straight line such that its acceleration at time 𝑑 seconds is given by π‘Ž = (2𝑑 βˆ’ 18) m/sΒ², 𝑑 β‰₯ 0. Given that its initial velocity is 20 m/s, and its initial displacement is 0 m, find an expression for the displacement of the particle at time 𝑑.

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Video Transcript

A particle is moving in a straight line such that its acceleration at time 𝑑 seconds is given by π‘Ž equals two 𝑑 minus 18 meters per seconds squared, where 𝑑 is greater than or equal to zero. Given that its initial velocity is 20 meters per second and its initial displacement is zero meters, find an expression for the displacement of the particle at time 𝑑.

Remember, acceleration is the rate of change of the velocity of an object. This means we differentiate the function for velocity to find the function for acceleration. We can conversely say that the integral of the function for acceleration will provide us with the function for velocity. Similarly, velocity is equal to the derivative of displacement with respect to time. So we can say that to find displacement, we can integrate the function for velocity with respect to time.

To answer this question then, we’re going to need to integrate our function for acceleration with respect to time twice. Throughout this process, we’ll use the fact that we know its initial velocity and its initial displacement. This will help us to find a particular expression for displacement. So velocity is the indefinite integral of two 𝑑 minus 18 with respect to 𝑑. The integral of two 𝑑 is two 𝑑 squared over two. The integral of negative 18 is negative 18𝑑. And of course, we mustn’t forget that we have this constant of integration 𝑐. And we see that 𝑣 is equal to 𝑑 squared minus 18𝑑 plus 𝑐. This is known as the general expression. We, however, know that its initial velocity is 20 meters per second. So we can say that when 𝑑 is equal to zero, 𝑣 is equal to 20.

And we can use this information to find the particular expression for velocity. We substitute 𝑑 equals zero and 𝑣 equals 20 into this equation. And we obtain 20 equals zero squared minus 18 times zero plus 𝑐, which gives us 20 equals 𝑐. And we have the expression for velocity at time 𝑑. It’s 𝑑 squared minus 18𝑑 plus 20. We’re gonna integrate one more time to find the expression for displacement. It’s the integral of 𝑑 squared minus 18𝑑 plus 20 with respect to 𝑑.

This time, the integral of 𝑑 squared is 𝑑 cubed over three. The integral of negative 18𝑑 is negative 18𝑑 squared over two. The integral of 20 is 20𝑑 and we have a constant of integration. Notice I’ve called this 𝑑 instead of 𝑐 because we’ve already used 𝑐 in this question. So 𝑠 is equal to 𝑑 cubed over three minus nine 𝑑 squared plus 20𝑑 plus 𝑑. Once again, we have enough information to work out the value of 𝑑. We know the initial displacement is zero meters. So when 𝑑 is equal to zero, 𝑠 is equal to zero. And we substitute these values in. And we obtain zero equals zero cubed over three minus nine times zero squared plus 20 times zero plus 𝑑, which tells us that zero is equal to 𝑑. And we’re done! We found the expression for the displacement of the particle at time 𝑑. It’s 𝑑 cubed over three minus nine 𝑑 squared plus 20𝑑 meters.

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