An object starts from rest and accelerates along a nine-meter-long straight line at a rate of two meters per second squared. What final velocity does the object have?
Alright, so in this question, the first thing that we’ve been told is that we’ve got an object which starts from rest. In other words, the initial velocity which we’ll call 𝑢 of the object happens to be 𝑢 is equal to zero meters per second because the object is at rest; it’s stationary, it’s not moving.
Now, secondly, we’ve been told that the object accelerates along a nine-meter-long straight line. In other words, if the object starts here, then let’s say it accelerates along a straight line, where this line has a length of nine meters. In other words then, the displacement of the object — the distance travelled in a straight line — which we’ll call 𝑠 happens to be 𝑠 is equal to nine meters.
And we know that the object is accelerating at a rate of two meters per second squared, that means that the object is speeding up. In other words, every second, the object is getting two meters per second of velocity. So let’s say that the acceleration which we’ll call 𝑎 of the object is 𝑎 is equal to two meters per second squared.
Now, lastly, we’ve been asked to find the final velocity of the object. Let’s call this final velocity 𝑣. And so we say that 𝑣 is equal to question mark cause that’s what we’re trying to find out. Now, in order to work out the value of 𝑣, we need to look at an equation that connects all of these quantities together: the initial velocity of the object, the displacement of the object, the acceleration of the object, and the final velocity of the object.
In this case, the equation we’re looking for is 𝑣 squared is equal to 𝑢 squared plus two 𝑎𝑠. In other words, the final velocity of the object squared is equal to the initial velocity of the object squared plus two times the acceleration of the object times the displacement of the object.
Now, in this case, we’re simply trying to solve for 𝑣. And we can do this by taking the square root of both sides of the equation. When we do this, on the left-hand side, the square in 𝑣 squared cancels with the square root. And so we’re left with 𝑣 is equal to the square root of 𝑢 squared plus two 𝑎𝑠.
Now, at this point, we can just substitute in our values. And before we do that, we can actually see that all the values we’ve been given are in their standard units. The initial velocity is in meters per second, standard unit. The displacement is in meters, again standard unit. The acceleration is in meters per second squared, standard unit once again. And so our final answer 𝑣 is going to be in its standard unit.
Now, 𝑣 is the velocity. So the unit is going to be meters per second. Knowing this, we can sub in our values. 𝑣 is equal to 𝑢 squared, that’s zero squared, plus two times 𝑎, that’s two, times 𝑠 that’s nine. And when we evaluate the right-hand side of this equation, we know that our final answer is going to be in meters per second.
Now, remember that on the right-hand side of the equation, we had a square root. Therefore, the answer that we get when we evaluate the right-hand side is going to be plus or minus six meters per second. So let’s consider the plus and the minus options separately.
First of all, the plus option, what this tells us is that the object starts out over here at rest, accelerates in this direction at two meters per second squared for a total distance of nine meters, and at the end, it’s now moving in the same direction — in the positive direction — at six meters per second.
This makes sense the object started out at rest, accelerated for a certain distance, and it’s now moving in the same direction as the acceleration at six meters per second.
The other option though, the negative option, doesn’t make quite as much sense because what we saying in that case is that the object starts at rest — at zero meters per second, accelerates in this direction, again arbitrary, but let’s say that this is the direction that accelerates in at two meters per second squared for nine meters, and then ends up at its final position, now travelling in the negative direction at six meters per second.
What! That doesn’t really add up. How can an object start out here, accelerate in this direction, and end up moving this way? So this second answer, the negative answer, is unphysical and we ignore it. Hence, we just take the positive root.
And then, we can say that the final velocity of the object is six meters per second.