# Racing Problem (Kinematics)

## 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.

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gneill
Mentor
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.

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
Mentor
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
PeroK
Homework Helper
Gold Member
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.

Last edited:
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!