A 0.25 kg puck is initially stationary on an ice surface...

A 0.25 kg puck is initially stationary on an ice surface with negligible friction. At time $t=0,$ a horizontal force begins to move the puck. The force is given by $\vec{F}=\left(12.0-3.00 t^{2}\right) \hat{\mathrm{i}}$ , with $\vec{F}$ in newtons and $t$ in seconds, and it acts until its magnitude is zero. (a) What is the magnitude of the impulse on the puck from the force between $t=0.500 \mathrm{s}$ and $t=1.25 \mathrm{s?}$ (b) What is the change in momentum of the puck between $t=0$ and the instant at which $F=0 ?$

A 0.25 kg puck is initially stationary on an ice surface with negligible friction. At time $t=0,$ a horizontal force begins to
move the puck. The force is given by $\vec{F}=\left(12.0-3.00 t^{2}\right) \hat{\mathrm{i}}$ , with $\vec{F}$
in newtons and $t$ in seconds, and it acts until its magnitude is
zero. (a) What is the magnitude of the impulse on the puck from the force between $t=0.500 \mathrm{s}$ and $t=1.25 \mathrm{s?}$ (b) What is the
change in momentum of the puck between $t=0$ and the instant
at which $F=0 ?$

Key Concepts

Momentum

Momentum is a fundamental physical quantity defined as the product of an object's mass and its velocity. It provides a measure of the object’s motion and is conserved in isolated systems. Understanding momentum is crucial for analyzing the effects of forces, as changes in momentum are directly linked to the impulses acting on an object.

Time Integration of Force

When a force varies with time, calculating the impulse requires the integration of the force function over the relevant time interval. This process effectively sums the contributions of the force at each instant, yielding the total impulse imparted, which in turn relates to the overall change in momentum.

Impulse

Impulse is a measure of the effect of a force acting over a period of time. It is defined as the product of the force and the time interval over which it is applied, and in cases where the force varies with time, it is calculated by integrating the force over that time interval. This concept is important because impulse directly quantifies the change in momentum caused by a force.

Impulse?Momentum Theorem

The impulse-momentum theorem states that the impulse acting on an object is equal to the change in its momentum. This principle is central to dynamics, as it connects the net effect of applied forces over time with the resulting change in an object’s state of motion, offering a powerful tool for analyzing motion in situations where forces are not constant.

Transcript

00:01 Our question says that a 0 .25 kilogram puck, i label that as m, is initially stationary on an ice surface with a negligible friction.

00:10 At times t equals zero, a horizontal force begins to move the puck.

00:14 And it says that the force is given by this expression here, which is 12 minus 3t squared, acting in the i -hat direction.

00:24 So i can put i -hat over here to indicate that this vector is acting in the i -hat direction.

00:30 Okay? now, the question says, a, what is the magnitude of the impulse on the puck between 0 .5 seconds and 1 .25 seconds.

00:41 And then for part b, what is the change momentum of the puck between t equals zero and the instance at which n? okay, so for part a, we can use the expression for the impulse, which says that j is equal to the integral of the force.

01:04 Which is a function of time, right? the time interval of the force as a function of time from some time, we'll call it t1 to t2.

01:22 Okay? well, for us, t1 is 0 .5 .00 seconds, and t2 is 1 .25 seconds.

01:37 1 .25, okay.

01:43 And the force here is 12 .0 minus 3 .00 t squared d t.

02:02 Okay.

02:03 So if we integrate this with respect to t, this is 12 .0t minus 1 3 .00 t to the third.

02:23 So if you take the time derivative of that, we.

02:25 Get back to our original expression for f.

02:28 And this entire thing is evaluated over the time interval of 0 .5 .00 seconds to 1 .25 seconds.

02:48 Okay, so first you plug in 1 .25 into 12t minus 1 3t to the third...