A race car moves such that its position fits
the relationship x=(5m/s) t + (.75m/s3 ) t3 where x is measured in meters and t
in seconds. A)Plot a graph of position versus time. B)Determine the
instantaneous velocity at t=4s using time intervals of .4s, .2s, and .1s.
I'm not looking for any answers, just a way to start B. For some reason, it's not clicking and it's not hard at all. Thanks ahead of time.
B) The velocity at time t is the derivative of the x(t) function. Therefore
V(t) = 5 + 2.25 t^2. You can use that formula for the exact instantaneous velocity. At t=4 s, the instantaneous velocity is 5 + 2.25*16 = 41 m/s.
They are NOT asking you to use differential calculus, but instead to compute the position at x= t and t + delta t, and divide by the interval delta t. For delta t = 0.1 s, you get
V = [x(4 +.1) - x(4)]/(0.1 s)
= (72.19 - 68.00)/0.1 = 41.9 m/s
The velocity will be the slope of the posistion-time graph. Draw a tangent to the curve at the points desidred, and draw a triangle , then determine the slope (delta position/deltatime).
No problem! Let's start by understanding the concept of instantaneous velocity. Instantaneous velocity is the velocity of an object at a specific point in time. In this case, we want to find the instantaneous velocity at t=4s.
To calculate the instantaneous velocity, we need to find the derivative of the position equation with respect to time (dt). The derivative of x(t) with respect to t gives us the velocity function v(t).
In this case, the position function is x(t) = (5m/s)t + (0.75m/s3)t³.
To find the derivative, you can follow these steps:
1. Differentiate each term individually. The derivative of (5m/s)t is 5m/s, and the derivative of (0.75m/s³)t³ is (2.25m/s³)t².
2. Combine the derivatives to get the velocity function v(t). In this case, v(t) = 5m/s + (2.25m/s³)t².
Once you have the velocity function v(t) = 5m/s + (2.25m/s³)t², you can find the instantaneous velocity at t=4s by plugging in the value of t into the velocity equation.
Now, let's calculate the instantaneous velocity at t=4s using different time intervals.
1. For the time interval of 0.4s: Plug t=4s+0.4s into the velocity equation and calculate the velocity.
2. For the time interval of 0.2s: Plug t=4s+0.2s into the velocity equation and calculate the velocity.
3. For the time interval of 0.1s: Plug t=4s+0.1s into the velocity equation and calculate the velocity.
By calculating the velocities for different time intervals, you can estimate the instantaneous velocity at t=4s. Remember, the smaller the time interval, the closer your estimate will be to the true instantaneous velocity.
I hope this explanation helps you get started on part B! Let me know if you have any further questions.