At the instant the traffic light turns green, a car that has...

At the instant the traffic light turns green, a car that has been waiting at an intersection starts ahead with a constant acceleration of 3.20 $\mathrm{m} / \mathrm{s}^{2} .$ At the same instant a truck, traveling with a constant speed of 20.0 $\mathrm{m} / \mathrm{s}$ , overtakes and passes the car. (a) How far beyond its starting point does the car overtake the truck? (b) How fast is the car traveling when it overtakes the truck? (c) Sketch an $x$ -t graph of the motion of both vehicles. Take $x=0$ at the inter-section. $\left(\text { d) Sketch a } v_{x}-t \text { graph of the motion of both vehicles. }\right.$

Key Concepts

Relative Motion

Relative motion examines the positions and speeds of different objects as they move in relation to one another. When comparing an accelerating car to a truck moving at a constant speed, it is essential to determine the point at which both objects share the same position, which requires setting up equations that account for each object's motion characteristics.

Graphical Analysis in Kinematics

Graphical analysis involves plotting and interpreting position versus time (x-t) and velocity versus time (v-t) graphs to visually understand the motion of objects. The x-t graph shows trajectories and intersection points, indicating where one object overtakes another, while the v-t graph illustrates how speed changes over time for objects with constant acceleration and constant speed.

Constant Acceleration Kinematics

This concept involves movement where acceleration is uniform over time, allowing the use of specific equations that relate initial velocity, final velocity, displacement, time, and acceleration. In problems like this one, where an object accelerates from rest with a constant acceleration, these equations provide a systematic method to compute distances covered and speeds reached after a given time.

Uniform Motion

Uniform motion refers to the movement of an object at a constant speed and in a straight line, which implies that its displacement is directly proportional to time. In this context, the truck is an example of uniform motion, making it straightforward to calculate its position at any given time using the linear relationship between distance and time.

Transcript

00:01 For this question, we have a car that is accelerating from rest at the intersection, at the rate of 3 .2 meters per second square.

00:13 And at the same time, we have a truck that is at the same point, not accelerating, but moving off with a velocity of 20 meters per second.

00:26 So the truck of course will be accelerate pass and overtake the car at first.

00:38 We want to find when the car will move and start to overtake the truck.

00:48 So beyond that, at that point when it overtakes, at this second time, when it overtakes, it is when both the car and the truck actually traveled the same amount of distance, right? this unknown distance over here and also at the same amount of time t.

01:20 So for our truck, the distance d is just 20 meters plus seconds time t times t.

01:30 Whereas for the car, this distance is equals to ut plus half a t square where u is the initial velocity where initial velocity is 0 so we don't have this term we just have half a t square it's just half times 3 .2 times t squared and since they both have traveled the same distance we can equate these two equations together and actually we can quickly solve for t which is 12 .5 seconds and therefore we can find this distance d by just putting in 12 .5 into here so we get 20 times 12 .5 to give us 250 meters now for the next part you'll find how fast the car is traveling when it just overtakes.

02:55 So given that it's constantly accelerating at this rate, so the final velocity will be just aat since it has no initial velocity.

03:04 So the total time is 2 .5, so just 2 .5 seconds times 3 .2 meters per second square and we would have speed of about 40 meters per second.

03:29 Now to sketch the graph for both vehicles.

03:40 I'm gonna start off with the position time graph.

03:47 We start off at the intersection being at the origin over here.

03:53 Now for the truck, it's quite simple because it is constantly moving at a constant velocity...

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