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A car's velocity as a function of time is given by $v_x(t) = \alpha + \beta t^2$, where $\alpha =$ 3.00 m/s and $\beta =$ 0.100 m/s$^3$. (a) Calculate the average acceleration for the time interval $t =$ 0 to $t =$ 5.00 s. (b) Calculate the instantaneous acceleration for $t =$ 0 and $t =$ 5.00 s. (c) Draw $v_x-t$ and $a_x-t$ graphs for the car's motion between $t =$ 0 and $t =$ 5.00 s.
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Physics 101 Mechanics
Chapter 2
Motion Along a Straight Line
Section 3
Average and Instantaneous Acceleration
Applying Newton's Laws
University of Michigan - Ann Arbor
University of Washington
University of Sheffield
Lectures
03:43
In physics, dynamics is the branch of physics concerned with the study of forces and their effect on matter, commonly in the context of motion. In everyday usage, "dynamics" usually refers to a set of laws that describe the motion of bodies under the action of a system of forces. The motion of a body is described by its position and its velocity as the time value varies. The science of dynamics can be subdivided into, Dynamics of a rigid body, which deals with the motion of a rigid body in the frame of reference where it is considered to be a rigid body. Dynamics of a continuum, which deals with the motion of a continuous system, in the frame of reference where the system is considered to be a continuum.
04:34
In physics, kinematics is the description of the motion of objects and systems in the frame of reference defined by the observer. An observer has to be specified, otherwise the term is meaningless.
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it turns out in doing part of this problem you'LL actually end up doing party as well Party ask for the average exploration between time is equal five seconds and time equals zero seconds. The formula for the average acceleration would be the acceleration at five seconds in this case minus the exploration at zero seconds over the time difference which is just five. My serum the figure out what these values are meaning to differentiate the velocity of suspects. Time to get the acceleration expects and we get the offer goes away and we get to bait a times time. Now we can figure out what the values are. A five is equal to ten beta and then play in our value of beta, which is point one. We just get one meter per second squared a zero is he called a zero meters per second squared. These are actually the answers to part B of the problem, but we need them to calculate part of the problem. We can just put in these and we get one minus zero over five and so the average acceleration for the trip is one fifth or point to meters per second squared and so that's parts A and B Down part C asked for grafts of Vienna versus T for time ranging from zero to five seconds and so we can start with the velocity. So if this is B and honesty, we know that it's going to have a parabolic shape because of this form. Alfa plus beta t squared. If we plug in, tea is equal to zero, we get that the velocity is equal to Alfa. So this point here is Alfa Alfa's three meters per second. So that's three million per second. At this point, it's going to curb up like a standard problem until it reaches by seconds. And that's the only range that we really care about. At this point, we can figure out the new velocity value by going back here employing Auntie's Eagle five seconds. If we square five, we get twenty five, twenty five times beta plus Alfa ends up being five point five years per second and so this Y value is five point five years per second. In orderto calculate the acceleration we refer back to this A of T is equal to two ability, so a T is equal to zero is equal to zero. And so this is point on the graph, and we only care about zero five. So I'm going to go and put in five for my tea. I should write a here, so when he is equal, five is equal to ten times beta, which is equal to one meter per second squared. So if I put that point that is one meter per second squared to get the points in between, we just simply note that this is a linear line. So it's just going straight. And so those are the two graphs, one of the vs T and any other of A versus team. One's a proble, the others just a standard line. And you'LL know that this is the derivative of this problem, which makes sense with what? The salaciousness with respect to velocity
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