Question Video: Using the Net Change Theorem Mathematics

Video Transcript

A particle accelerates at the rate of two 𝑡 plus seven meters per second squared after 𝑡 seconds of motion. If 𝑣 evaluated at zero is equal to negative eight meters per second, how long does it take for the velocity to reach 50 meters per second? Give your answer to two decimal places.

The question tells us that a particle is accelerating at a rate of two 𝑡 plus seven meters per second squared. We’re told that 𝑣 evaluated at zero is equal to negative eight meters per second. That means the initial velocity of our particle is negative eight meters per second. The question wants us to use this information to find how long it takes for the velocity of our particle to reach 50 meters per second. It wants us to give our answer to two decimal places.

To start, we know the acceleration function of our particle is given by two 𝑡 plus seven. The question wants us to find the time taken for our particle to reach a certain velocity. So we want to find an equation for the velocity of our particle. To do this, we recall the acceleration of a particle is equal to the rate of change in velocity with respect to time. The converse of this statement is also true. The velocity of our particle is equal to the integral of the acceleration with respect to time. And this will be true up to a constant of integration.

Using this, we have the velocity of our particle after 𝑡 seconds is given by the integral of two 𝑡 plus seven with respect to 𝑡. We can evaluate this integral by using the power rule for integration which tells us, for constants 𝑎 and 𝑛, where 𝑛 is not equal to negative one. To integrate 𝑘 times 𝑡 to the 𝑛th power with respect to 𝑡, we add one to our exponent and then divide by this new exponent. Then, we add our constant of integration. It might help us to think of the constant seven as seven times 𝑡 to the zeroth power. Applying the power rule of integral to each term in our integrand gives us two 𝑡 squared over two plus seven 𝑡 divided by one plus 𝑐.

And we can simplify this. Two divided by two simplifies to give us one, and seven divided by one simplifies to give us seven. So we’ve shown that the velocity of our particle is given by 𝑡 squared plus seven 𝑡 plus 𝑐. We actually call this a general solution because we don’t know the value of the constant 𝑐. We can actually find the value of 𝑐 in this case. The question tells us that 𝑣 evaluated at zero is equal to negative eight meters per second. In other words, when we substitute 𝑡 is equal to zero into our velocity function, we should get the output negative eight.

Substituting 𝑡 is equal to zero into our equation for the velocity gives us negative eight is equal to zero squared plus seven times zero plus 𝑐. And we see that zero squared and seven times zero are both equal to zero. So we’ve shown that 𝑐 is equal to negative eight. Then, substituting this value of 𝑐 into our general solution to the velocity function, we get that 𝑣 of 𝑡 is equal to 𝑡 squared plus seven 𝑡 minus eight. And this function tells us the velocity of our particle after 𝑡 seconds. And the question wants us to find the value of 𝑡 where the velocity of our particle is equal to 50 meters per second. This means we just need to solve our velocity function equal to 50.

So we want to find the values of 𝑡 where 50 is equal to 𝑡 squared plus seven 𝑡 minus eight because then the velocity of our particle will be 50 meters per second. We can then subtract 50 from both sides of this equation. And we get the quadratic zero is equal to 𝑡 squared plus seven 𝑡 minus 58. We could solve this by using the quadratic formula or we could just use our calculator. Doing this gives us the following two answers to two decimal places, 4.88 and negative 11.88.

And remember, we’re told that our acceleration function is valid after 𝑡 seconds of motion. This tells us that 𝑡 is greater than or equal to zero, so we can’t have a negative value of 𝑡. Therefore, our answer will be 4.88. And all of our units are given in terms of meters and seconds. So since this just represents the time it takes for the velocity to reach 50 meters per second, we can add the units of seconds.

Therefore, we’ve shown if a particle is accelerating at a rate of two 𝑡 plus seven meters per second squared after 𝑡 seconds of motion. And the particle’s initial velocity is negative eight meters per second, then to two decimal places, it will take 4.88 seconds for the particle to reach a velocity of 50 meters per second.