A 5.0 kg toy car can move along an x axis; Fig. 9-50 gives...

A $5.0 \mathrm{~kg}$ toy car can move along an $x$ axis; Fig. $9-50$ gives $F_{x}$ of the force acting on the car, which begins at rest at time $t=0$. The scale on the $F_{x}$ axis is set by $F_{x z}=5.0 \mathrm{~N}$. In unit-vector notation, what is $\vec{p}$ at (a) $t=4.0 \mathrm{~s}$ and $(\mathrm{b}) t=7.0 \mathrm{~s}$, and $(\mathrm{c})$ what is $\vec{v}$ at $t=9.0 \mathrm{~s}$ ?

Key Concepts

Impulse-Momentum Theorem

This theorem states that the impulse applied to an object, which is the integral of force with respect to time, causes a change in the object’s momentum. It is a fundamental principle used to relate force functions over time to the resulting changes in momentum.

Integration of Force

Integrating the force over a time interval gives the impulse delivered to an object. This integration is crucial for problems where the force varies with time, as it allows one to compute the change in the object's momentum directly from the force's time dependence.

Newton’s Second Law

Newton's Second Law connects the net force acting on an object to its acceleration through the object’s mass. This relationship forms the basis for linking force, acceleration, and momentum, especially when analyzing motion resulting from varying forces.

Unit Vector Notation

Unit vector notation is used to represent vector quantities like force, momentum, and velocity in a clear and standard way by showing both magnitude and directional components. This notation is essential for describing motion along specified axes in physics.

Relationship Between Momentum and Velocity

The momentum of an object is defined as the product of its mass and velocity. This relationship is a cornerstone in kinematics and dynamics, allowing one to infer an object's velocity from its momentum or vice versa, assuming the mass is known.

Transcript

00:03 So for this problem, i've drawn a little bit, a little graph to mimic the one that's in the book.

00:08 And so it's a little bit crude, but it's okay.

00:13 And so we have one thing given to us.

00:17 Well, we have a few things given to us.

00:18 We have this five right here.

00:22 And so that means we know that this is 10, and we know that this is negative 5.

00:26 And we have the mass of the cart, which is 5 kilograms.

00:31 So what we're going to do is we need.

00:33 The first part of the question asks us to find the momentum at time.

00:36 Equals four seconds.

00:38 So what we're going to do is we're going to use this equation, change in momentum equals force times the change in time.

00:48 But since the force is not constant, we're going to have to integrate.

00:51 So we have momentum equals force d t.

01:01 So what we can do is we can just take the area under the curve because we've got some nice straight lines, some rectangles and some triangles to find this momentum.

01:12 So what we're going to do is we're going to add up the area under the curve here and here until we get to 4.

01:20 So we'll take the area of this triangle and on this rectangle.

01:25 So let's do that.

01:26 We get the momentum equals 10 times 2 divided by 2.

01:33 So the triangle plus 10 times and that turns out to be 30.

01:42 So that is our momentum for part a...

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