A force of magnitude Fx acting in the x-direction on a 2.55-kg particle varies in time as shown in the figure below. (Indicate the direction with the sign of your answer.)

(a) Find the impulse of the force. (Give your answer to one decimal place.)
kg · m/s

(b) Find the final velocity of the particle if it is initially at rest.
m/s

(c) Find the final velocity of the particle if it is initially moving along the x-axis with a velocity of
−2.10 m/s.

m/s

swag

To answer these questions, we'll need the figure that shows how the force varies in time. However, I can still explain the steps you need to follow to solve these problems.

(a) The impulse of a force is defined as the change in momentum that the force produces. To find the impulse, you need to integrate the force over time. The impulse is given by the equation:
Impulse = ∫ F(t) dt
where F(t) is the force as a function of time, and the integral is taken over the time interval shown in the figure. The result will be in units of kg·m/s.

(b) To find the final velocity of the particle if it is initially at rest, you can use the impulse-momentum theorem. According to the theorem, the impulse applied to an object is equal to the change in its momentum. So, you can calculate the impulse from part (a) and then use it to find the final velocity using the equation:
Impulse = Change in Momentum
Impulse = m * (final velocity - initial velocity)
where m is the mass of the particle. Rearranging this equation will allow you to solve for the final velocity.

(c) To find the final velocity of the particle if it is initially moving along the x-axis with a velocity of -2.10 m/s, you can use a similar approach as in part (b). Again, calculate the impulse from part (a), and then use it to find the final velocity using the equation:
Impulse = Change in Momentum
Impulse = m * (final velocity - initial velocity)
where m is the mass of the particle and the initial velocity is -2.10 m/s. Rearrange this equation to solve for the final velocity.

I hope this helps guide you through the solution even without access to the figure.

To solve these problems, we need to use the concept of impulse and the relationship between force, mass, and acceleration. Here's how you can find the answers:

(a) To find the impulse of the force, we need to calculate the area under the force-time graph.

First, determine the time interval for which the force is acting. In the given graph, the force is acting from t = 0s to t = 2s.

Next, calculate the area under the graph using the equation for impulse:

Impulse = ∫F(t) dt

Since the force is constant, the integral can be simplified to:

Impulse = F * ∫dt

Integrating from t = 0s to t = 2s:

Impulse = F * (t)|[0,2]

Impulse = F * (2 - 0)

Therefore, the impulse of the force is equal to F times the time interval for which it acts.

(b) To find the final velocity of the particle if it is initially at rest, we can use the concept of impulse-momentum theorem. The impulse acting on a particle is equal to the change in momentum.

Impulse = Change in Momentum

Since the particle is initially at rest, its initial momentum is zero.

Impulse = Final Momentum - Initial Momentum

Therefore, the impulse of the force is equal to the final momentum of the particle.

We can calculate the impulse from part (a) and use it to find the final velocity.

Impulse = Mass * Final Velocity

Rearranging the equation, we get:

Final Velocity = Impulse / Mass

Substitute the values of impulse (from part (a)) and mass (given as 2.55 kg) into the equation to calculate the final velocity.

(c) To find the final velocity of the particle if it is initially moving along the x-axis with a velocity of -2.10 m/s, we can use the same concept as part (b) but consider the initial velocity.

Impulse = Change in Momentum

Impulse = Final Momentum - Initial Momentum

Impulse = Mass * Final Velocity - Mass * Initial Velocity

Now, rearrange the equation and solve for the final velocity:

Final Velocity = (Impulse + Mass * Initial Velocity) / Mass

Substitute the values of impulse (from part (a)), mass (given as 2.55 kg), and initial velocity (-2.10 m/s) into the equation to calculate the final velocity.

I can not see your graph but use the logic from the other problem I just helped you with.

F = m a = m dv/dt

impulse = integral F dt = final mv - initial m v