HINTS FOR PROBLEM SET #1

Problem Set #1 – Problem 1

The motion of a particle in rectilinear motion is identified by one position coordinate.

The problem statement gives a relation x in millimeters as a function of t in seconds.

You must conclude that x is a position coordinate and that this is rectilinear motion.

So to determine the position of the particle at t = 3sec, just plug t = 3 into the expression given for x.

Now you want to determine the velocity of the particle at a specific instant in time.

So you need to find an expression of velocity as a function of time, v(t).

Take the derivative of x(t) with respect to time.

This gives v(t), the expression for velocity of the particle as a function of time. 

So to determine the velocity of the particle at t = 3sec, just plug t = 3 into the expression for v(t).

Lastly, you want to determine the acceleration of the particle at a specific instant in time.

So you need to find an expression of acceleration as a function of time, a(t).

Take the derivative of v(t) with respect to time.

This gives a(t), the expression for acceleration of the particle as a function of time. 

So to determine the acceleration of the particle at t = 3sec, just plug t = 3 into the expression for a(t).

Problem Set #1 – Problem 2

You want to find the velocity of a particle at a specific instant in time.

So you need to find an expression for velocity as a function of time, v(t).

Use the fact that a = dv/dt and the given fact a = kt. (k is a constant of proportionality.)

So kt = dv/dt.

Now separate the variables and integrate using one of the given boundary conditions.

(The boundary condition you should use is the given initial condition:

The fact that at t = 0, v = 16in/sec.)

If you use the definite integral:

     On the dt side you will be integrating between the limits of 0sec and t.

     On the dv side you will be integrating between the limits of 16in/sec and v.

Otherwise you can plug in the initial condition to obtain the constant of integration.

You then find k, the proportionality constant, by using the other boundary condition.

(The fact that at t = 1sec the velocity = 15in/s.)

Now you have the expression of velocity as a function of time, v(t).

Plug t = 7sec into this expression to get the velocity at 7sec.

Now you want to find the position of a particle at a specific instant in time.

So you need to find an expression of position as a function of time, x(t).

You just got velocity as a function of time, v(t).

So use the fact that v = dx/dt. and replace v with the expression for v(t).

Then separate the variables and integrate using the given boundary condition.

For a definite integral:

     The limits on the dt side will be from 1sec to t.

     The limits on the dx side will be from 20inches to x.

Now you have the expression of position as a function of time, x(t). 

Plug t = 7sec into this expression to get the position at 7sec.

Lastly, you want to find the total distance traveled from t = 0sec to t = 7sec.

You might be tempted to just plug t = 7sec into the expression for x(t).

But this will only give the position at t = 7sec, not the total distance traveled.

The position and the total distance traveled are not the same thing!

What if the particle retraced its steps?

Then the distance traveled will be greater than the distance from the origin.

So you need to determine if the velocity changed direction at any time.

How? Read on!

Go back to the expression for v(t).

At an instant when the velocity changes direction, it will be zero.

So to find those instants plug v = 0 into the expression for v(t) and solve for t.

You should get the answer of t = 4sec.

Find the position at t = 0, t = 4 and t = 7.

The total distance traveled will be |x(4) – x(0)| + |x(7) – x(4)|

Problem Set #1 – Problem 3

Part a:

You want to find the velocity of the particle at a specific position.

So you need to find an expression for velocity as a function of position, v(x).

Use the fact that a = v dv/dx and the given fact that a = -k v^1/2.

So -k v^1/2 = v dv/dx.

Separate the variables and integrate using the boundary conditions.

You should have the expression for v(x).

Plug x = 20m into this expression to get the velocity of the particle at the position of 20m.

Part b:

You want to find the time required for the particle to come to rest.

So you need to find the expression of velocity as a function of time, v(t).

Then you will set velocity to zero and solve for t.

To get v(t) use the fact that a = dv/dt and the given fact that a = -k v^1/2.

So -k v^1/2 = dv/dt. (You already found k.)

Separate the variables and integrate using the boundary condition.

Now you should have the expression for v(t).

Plug 0in/sec for v and solve for t.