# Physics Exercises/Kinematics in One Dimension/Solutions

## Without CalculusEdit

**3.1 s**.Use the kinematics equation . Let the rock's initial height . Then equation simplifies to . So, two solutions for

*t*are:*t = 0*s and . Plugging in and gives*t = 3.1 s*.- Let
*L*be the length of the escalator,*v*be the speed of the escalator, and_{1}*v*be the speed the man walks on solid ground. It takes_{2}*t*for a man standing on the escalator to travel its length, so . If the man walks against the escalator, his net speed is ; it takes him_{1}*t*to travel the length of the escalator, so . Since , we find that or . Now, when the man walks down the escalator, his net speed is . If it takes him_{2}*t*seconds this time, then . Rewriting in terms of given quantities, , and solving for*t*, we get:. Note that by carrying out the calculation in symbols as far as we can, we actually discover that the final answer is independent of

*L*. This is one of many benefits of*not*substituting in numbers until the very end. (As with any rule, this rule has exceptions, but start by following the rule—you will notice exceptions by yourself when you become sufficiently skilled.) **14.4 s**and**200 m**Let

*x*be the position of the bike and_{b}*x*be the position of the car. We know from our kinematic equations that . Because we are assuming that both the bike and the car are moving at a constant velocity (25 km/hr and 50 km/hr respectively) we can say that the acceleration (a) in this equation is zero (in both cases). Thus we can write two equations of motion :_{c}The equation for the bike is and the equation for the car is .

Because we are interested in the point at which the two meet - we can safely say that we are looking for the point at which . Solving this expression gives us and thus the time which it will take for the two objects to reach the same location can be found to be . Before we can plug in any values we need to make sure that they share the same units, so 25 km/hr becomes 6.944 m/s and 50 km/hr becomes 13.888 m/s. Plugging in these values gives us

Finally, in order to find the location at which they meet, simply plug this value for time into either of the two equations of motion, and we get

**18 mph**

The speed of Cinderella, is 12mph

The time used by Cinderella, is 0.25h

The speed of Prince, is unknown

The time used by Prince, is 0.16666h or

Since they have to meet, their distance traveled will be equal, then

## With CalculusEdit

The basic equations are these: , . Since our final goal is

*x*as a function of*t*, we should solve for*v(t)*first by integrating the first expression:And integrating this a second time yields the final expression for

*x(t)*. .*e*cmThe positions as a function of time can be written down as the infinite series:

. Once you let , you should recognize the resulting series as the Taylor series of (times 1 cm):. If you plug in , you get .

Observant reader should notice that we suddenly dropped units in the middle. To be thorough, we would factor out "1 s" from each written expression of

*t*and absorb it into the coefficient (that way,*t*becomes unitless and all coefficients have unit of cm).