Velocity (avg/instantaneous)

[phragg]

A position-time graph for a particle moving along the x-axis is shown below:

(a) Find the average velocity in the time interval t = 1.50 s to t = 4.00 s.
(b) Determine the instantaneous velocity at t = 2.00 s by measuring the slope of the tangent line shown in the graph.
(c) At what value of t is the velocity zero?

http://img241.imageshack.us/img241/1514/tangetsb6.jpg

I figured that (a) was the easy part as I went ahead to solving that
$$v_{ave} = \frac{\Delta x}{\Delta t}$$

$$\Delta t = t_f - t_i$$
$$\Delta t = 4.00 s - 1.50s$$
$$\Delta t = 2.50 s$$

$$\Delta x = x_f - x_i$$
$$\Delta x = 2m - 7m$$
$$\Delta x = -5m$$

$$v_{ave} = \frac{-5m}{2.50s}$$
$$v_{ave} = -2m/s$$

So after that was done I went on to part (b) which first asked to find the slope of the tangent point was easily done by:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
$$m = \frac{0 - 12}{4 - 0}$$
$$m = \frac{-3}{1}$$

Now I am completely stumped as to what they're asking for how to Determine the instantaneous Velocity at t = 2.00 s and Don't get me started on (c). Oh an p.s. Hey I'm new to PF ;p!

 

[CompuChip]

Hi, welcome to PF.
Actually, you already solved b :)
The key thing you are missing, is that instantaneous velocity at some time is the slope of the (t, x) graph at that point.

This should also help you solve c.

 

[baba89]

Hello and welcome to PF! It's great to have you here. I can provide some insight into the questions you have raised about velocity.

First, let's clarify the difference between average velocity and instantaneous velocity. Average velocity is the overall rate of change of an object's position over a given time interval, while instantaneous velocity is the rate of change at a specific moment in time.

In part (a), you correctly calculated the average velocity for the given time interval. This tells us the overall rate of change of the object's position from t = 1.50 s to t = 4.00 s.

In part (b), you have correctly calculated the slope of the tangent line at t = 2.00 s. This slope represents the instantaneous velocity at that specific moment in time.

To determine the instantaneous velocity at t = 2.00 s, you simply need to plug in the values for x and t into the equation for velocity:
v = \frac{dx}{dt}
v = \frac{-3m}{1s}
v = -3m/s

This tells us that at t = 2.00 s, the object's instantaneous velocity is -3 m/s.

For part (c), we are looking for the value of t at which the velocity is zero. This means that the object is not moving, or it has stopped at that point. To find this, we can use the slope of the tangent line at t = 0 s and t = 4.00 s.

At t = 0 s, the slope of the tangent line is 0, which means the object is not moving. At t = 4.00 s, the slope of the tangent line is -3, which means the object is moving in the negative direction.

Therefore, the velocity is zero at some point between t = 0 s and t = 4.00 s. To find the exact value of t, we can use the equation for velocity and set it equal to zero:
v = \frac{dx}{dt} = 0
\frac{dx}{dt} = 0
This means that the object's position is not changing, or it is not moving, at that specific moment in time.

I hope this helps clarify the concept of average and instantaneous velocity. Keep up the good work!