A crate on a motorized cart starts from rest and moves with...

A crate on a motorized cart starts from rest and moves with a constant eastward acceleration of $a=2.80 \mathrm{m} / \mathrm{s}^{2}$ . A worker assists the cart by pushing on the crate with a force that is eastward and has magnitude that depends on time according to $F(t)=$ $(5.40 \mathrm{N} / \mathrm{s}) t .$ What is the instantaneous power supplied by this force at $t=5.00 \mathrm{s} ?$

Key Concepts

Constant Acceleration Kinematics

This concept involves using the equations of motion for an object starting from rest with constant acceleration. It allows us to determine the instantaneous velocity at any given time using the relationship v = at, which is essential when linking the object's motion with the force applied.

Time-Dependent Force

A force that varies with time means its magnitude is a function of time rather than being constant. Understanding how to work with such functions is crucial when computing instantaneous quantities like power, as the value of the force at a specific moment must be used.

Instantaneous Power Calculation

Instantaneous power is defined as the rate at which work is done at a specific moment, given by the dot product of force and velocity vectors. When force and velocity are aligned, power simplifies to the product of their magnitudes, making it necessary to obtain the value of velocity and the time-dependent force at the same instant.

Transcript

00:01 Once again welcome to a new problem.

00:02 This time we're going to make connections between power force on time s.

00:10 So we have a firing false, and therefore we always have to think about time.

00:15 We have a card.

00:18 It's a motorized cart that moves in the east west direction.

00:24 So this is, you know, it was left to have west and then towards the right, we have east.

00:30 So on top of the card, we have ah, crate and a zit.

00:38 There's a there's a locker that's pushing it.

00:42 So they're pushing it that way and saw the crate is moving with an acceleration or the to the great and the cart the moving with an acceleration of two point eight zero meters per second there.

00:55 The thing is that the force being applied is a variable force off five point four on newton for second andi and time is a parameter off that force.

01:08 So it's this dependent on time as a variable.

01:12 So we want to find the instantaneous power power instantaneous, uh, instantaneous power when the time is five seconds.

01:28 Remember, powell has a relationship between force and velocity.

01:32 These of actor quantities vector quantities have magnitude and direction.

01:37 And so the fast i'm going to do is to compute that factor force at five seconds and that becomes, um we'll use this formula right here.

01:50 So we'll plug in time.

01:51 So we have five point four zero newtons for second on then times the time, which is five seconds that uses on instantaneous force at five seconds off twenty seven mutants.

02:07 Beyond the force we need to compute the velocity so assumed this will sti equals to zero and it's travelling a specific distance up until t equals two five seconds and we're having a force off twenty seven newton's at that point.

02:23 Kinnah matic equations can help us determine the velocity off the great no and the cart at five seconds.

02:32 Member of the final equals two v initial of plus eighty.

02:37 The initial the last he won the cards starts happened to be zero meters per second, so this part will just zero and we want to find the final boss to take...

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