How Long Does it Take for a Motorcycle to Overtake a Car?

[rtareen]

Homework Statement

The motorcycle first takes the lead because its (constant) acceleration am = 8.40 m/s2 is greater than the car’s (constant) acceleration ac = 5.60 m/s2, but it soon loses to the car because it reaches its greatest speed vm = 58.8 m/s before the car reaches its greatest speed vc = 106 m/s. How long does the car take to reach the motorcycle?

Relevant Equations

ac = 5.6m/s2
am = 8.40m/s2
vcmax = 106m/s
vmmax = 58.8 m/s

Here is my attempt at the solution but I got the wrong answer. The right answer is t=16.6s. I know from the book (this is an example problem) that the motorcycle reaches its max speed at t=7.0s. But I don’t know where I made the mistake that is causing me to get the wrong answer afterwords.

 

[gneill]

Your error lies in how you calculate the motorcycle position.

You're covering the time between 0 s and 7.0 seconds twice, since your xmo already covered that once.

 

[rtareen]

Your error lies in how you calculate the motorcycle position.
View attachment 254545
You're covering the time between 0 s and 7.0 seconds twice, since your xmo already covered that once.

I don’t see why that’s a mistake. At t=7.0s the equation for the motorcycles position changes. It no longer has an acceleration, just a constant velocity. xmo is the starting point for this new position function.

 

[gneill]

xmo is the starting point for this new position function.

The way it is written means that the $v \times t$ term assumes a starting time of $t = 0$. That is not the case if it's contribution should only "kick in" at $t = 7 sec$.

 

[PeroK]

I don’t see why that’s a mistake. At t=7.0s the equation for the motorcycles position changes. It no longer has an acceleration, just a constant velocity. xmo is the starting point for this new position function.

It seems that your answer of $9.6s$ is the time starting from when the motorcycle reaches its maximum speed (after $7s$). The answer of $16.6s$ is the time from when both vehicles start. I.e. your answer plus the $7s$.

PS after $9.6s$ the speed of the car is $53.76 m/s$, so it's not yet reached the speed of the motorcycle.

 

[rtareen]

It seems that your answer of $9.6s$ is the time starting from when the motorcycle reaches its maximum speed (after $7s$). The answer of $16.6s$ is the time from when both vehicles start. I.e. your answer plus the $7s$.

PS after $9.6s$ the speed of the car is $53.76 m/s$, so it's not yet reached the speed of the motorcycle.

So I just need to add 7s to the answer I got and I’ll have the correct answer. I can see why now. Thanks!