Question Video: Determining the Velocity of a Particle based on Uniform Acceleration and Displacement Mathematics

Video Transcript

A particle is moving in a straight line such that its displacement 𝑠 after 𝑡 seconds is given by 𝑠 equals negative 10𝑡 cubed plus 12𝑡 squared plus 10𝑡 meters, for 𝑡 is greater than or equal to zero. Find the velocity of the particle when its acceleration is zero.

Here, we have an equation for 𝑠 in terms of 𝑡, where 𝑠 is the displacement of the particle. And so we recall that to link displacement and velocity, we differentiate our expression for displacement with respect to time such that 𝑣 is d𝑠 by d𝑡. We also know that acceleration is change in velocity with respect to time. So we differentiate our expression for velocity with respect to time. And that gives us an expression for acceleration. We could also say that since we’re differentiating a derivative, acceleration is the second derivative of 𝑠 with respect to time.

Now, we’re looking to find the velocity of the particle when its acceleration is equal to zero. So we’re going to find an expression for both velocity and acceleration then set our expression for acceleration equal to zero. By doing so, we’ll be able to find the time of which the acceleration is equal to zero. And then we can substitute this into our expression for velocity.

So let’s begin by differentiating 𝑠 with respect to 𝑡 term by term. We know that to differentiate a power term, we multiply the entire term by the exponent and then reduce that exponent by one. So the derivative of negative 10𝑡 cubed is negative 30𝑡 squared. Similarly, the derivative of 12𝑡 squared is two times 12𝑡, which is 24𝑡. Finally, the derivative of 10𝑡 is 10. So we have an equation for 𝑣. It’s negative 30𝑡 squared plus 24𝑡 plus 10. We’ll repeat this process, differentiating each of these terms with respect to 𝑡. That will give us an equation for 𝑎.

The derivative of negative 30𝑡 squared is two times negative 30𝑡; that’s negative 60𝑡. The derivative of 24𝑡 is 24. And when we differentiate a constant, we get zero. So the derivative of 10 is simply zero. Remember, we’re looking to find the velocity when the acceleration is equal to zero. So let’s set this equation equal to zero and solve for 𝑡. We add 60𝑡 to both sides to get 60𝑡 equals 24. Then we divide both sides of our equation by 60. So 𝑡 is 24 divided by 60, which simplifies to two-fifths or 0.4. So we can say that the acceleration of the particle is equal to zero at 𝑡 equals 0.4 seconds.

To find the velocity at this point, we simply substitute 𝑡 equals 0.4 into our function for velocity. That gives us negative 30 times 0.4 squared plus 24 times 0.4 plus 10, giving us a value of 14.8. Our displacement is measured in meters and our time in seconds. So the units for our velocity must be meters per second. The velocity of the particle when its acceleration is zero is 14.8 meters per second.