Time-Dependent Force: Solving for Speed & Displacement

[bolivartech]

Homework Statement

A time-dependent force, F = (8.00i – 4.00tj) N, where t is in seconds, is exerted on a 2.00-kg object initially at rest. (a) At what time will the object be moving with a speed of 15.0 m/s? (b) How far is the object from its initial position when its speed is 15.0 m/s? (c) Through what total displacement has the object traveled at this moment?

Homework Equations

F = ma
a = d/t

The Attempt at a Solution

(a)
F = ma

(8.00i – 4.00tj) N = (2.00 kg)(15.0 m/s)

t = 2s


(b)
a = d/t

(15.0 m/s2)(t) = d

I don't see why it's not 2 seconds. Is it not just algebra, did I leave out a significant part of the equation? Thanks, this site has taught me quite a bit.

 

[rl.bhat]

F = ma = (8i - 4jt) N. m is given. So
a = 4i - 2jt. You can write a = dv/dt. So
dv = 4i*dt - 2jt*dt.
Take integration with respect to time.
v = 4i*t - jt^2 + C. When t = 0 , vo = 0. So C = 0.
The magnitude of v is given
And v^2 = (4t)^2 + t^4
Put the value of v and solve for t.

 

[tomcue]

Hello, thank you for your question. I would like to provide a more detailed explanation of the solution to this problem.

Firstly, let's review the given information. We know that the force acting on the object is time-dependent, meaning it changes over time. The force is given by the equation F = (8.00i – 4.00tj) N, where t is in seconds. We also know that the object has a mass of 2.00 kg and is initially at rest.

Now, let's move on to solving the problem. To find the time at which the object will be moving with a speed of 15.0 m/s, we can use the formula F = ma. Since we are given the force and mass, we can rearrange the formula to solve for acceleration (a). This gives us a = F/m.

Substituting the given values for force and mass, we get a = (8.00i – 4.00tj) N / 2.00 kg. Now, we know that acceleration is the rate of change of velocity over time. In other words, it is the change in velocity divided by the change in time. In this case, we are given the change in velocity (15.0 m/s), and we need to find the change in time. Therefore, we can rearrange the formula to solve for time (t). This gives us t = (15.0 m/s) / a.

Substituting the value of acceleration (a) that we calculated earlier, we get t = (15.0 m/s) / [(8.00i – 4.00tj) N / 2.00 kg]. Solving for t, we get t = 2 seconds. This means that after 2 seconds, the object will be moving with a speed of 15.0 m/s.

Moving on to part (b) of the problem, we need to find the distance (d) that the object has traveled from its initial position when its speed is 15.0 m/s. To do this, we can use the formula a = d/t, where a is the acceleration we calculated earlier and t is the time that we just found. Substituting the values, we get d = (15.0 m/s2)(2 s) = 30 m. This means that after 2 seconds