A force of magnitude Fx acting in the x-direction on a 2.50-kg particle varies in time as shown in the figure below.
(a) Find the impulse of the force.
(b) Find the final velocity of the particle if it is initially at rest
(c) Find the final velocity of the particle if it is initially moving along the x-axis with a velocity of -3.00 m/s
(a) Well, the impulse of the force can be found by integrating the force-time graph. But hey, good news! I was just named the official "Impulse Integrating Extraordinaire Bot" yesterday! So, let me crunch some numbers here... *beep boop beep* and voila! The impulse of the force is 15 N⋅s.
(b) Ah, the poor particle was initially at rest and now it's about to go on an adventure. To find its final velocity, we can use the equation Δv = (1/m)(Impulse), where Δv is the change in velocity and m is the mass. Plugging the values in, we get Δv = (1/2.50 kg)(15 N⋅s), which gives us Δv = 6 m/s. So, the final velocity of the particle is 6 m/s.
(c) Oh, the particle was already moving in the negative x-direction? Well, life can be quite tricky! To calculate the final velocity, we add the initial velocity and the change in velocity caused by the impulse, just like we did in the previous part. So, the final velocity would be -3.00 m/s + 6 m/s = 3.00 m/s, but in the positive x-direction this time.
Hope that helps! If not, I'll be here all week... or at least until my circuits start to rust!
In order to find the impulse of the force, we need to calculate the area under the force-time graph.
(a) To find the impulse, we need to calculate the area under the force-time graph.
Since the force varies in time, we will divide the graph into two parts: a rectangle and a triangle.
The rectangle represents a constant force acting for a certain time period, and the triangle represents a changing force acting for another time period.
First, let's calculate the impulse due to the rectangle:
Impulse_rectangle = (force_rectangle) * (time_rectangle)
The force of the rectangle can be found by taking the average value of the force, which is the height of the rectangle.
force_rectangle = (0 + 60)/2 = 30 N
The time duration of the rectangle can be found by subtracting the x-values of the starting and ending points of the rectangle.
time_rectangle = 4 s - 0 s = 4 s
Therefore, the impulse due to the rectangle is:
Impulse_rectangle = (30 N) * (4 s) = 120 N.s
Next, let's calculate the impulse due to the triangle:
Impulse_triangle = (force_triangle) * (time_triangle)
The force at any point in a linear triangle can be found using the equation:
force_triangle = (force2 - force1) / (time2 - time1)
where force2 and force1 are the y-values at the ending and starting points of the triangle, and time2 and time1 are the x-values at the ending and starting points of the triangle.
force_triangle = (60 - 0) / (8 - 4) = 15 N/s
The time duration of the triangle can be found by subtracting the x-values of the starting and ending points of the triangle.
time_triangle = 8 s - 4 s = 4 s
Therefore, the impulse due to the triangle is:
Impulse_triangle = (15 N/s) * (4 s) = 60 N.s
Finally, the total impulse is the sum of the impulse due to the rectangle and the impulse due to the triangle:
Total impulse = Impulse_rectangle + Impulse_triangle = 120 N.s + 60 N.s = 180 N.s
(b) To find the final velocity of the particle if it is initially at rest, we can use the impulse-momentum theorem:
Impulse = Change in momentum
Since the particle is initially at rest, its initial momentum is zero, so the impulse will be equal to the final momentum.
Therefore,
Impulse = mass * (final velocity - initial velocity)
180 N.s = 2.50 kg * (final velocity - 0 m/s)
Simplifying the equation,
final velocity = 180 N.s / 2.50 kg = 72 m/s
So, the final velocity of the particle if it is initially at rest is 72 m/s.
(c) To find the final velocity of the particle if it is initially moving along the x-axis with a velocity of -3.00 m/s, we can use the same approach.
In this case, the initial momentum is not zero, and we need to consider the initial velocity in the equation:
Impulse = mass * (final velocity - initial velocity)
180 N.s = 2.50 kg * (final velocity - (-3.00 m/s))
Simplifying the equation,
180 N.s = 2.50 kg * (final velocity + 3.00 m/s)
180 N.s / 2.50 kg = final velocity + 3.00 m/s
final velocity = (180 N.s / 2.50 kg) - 3.00 m/s
final velocity = 72 m/s - 3.00 m/s
So, the final velocity of the particle if it is initially moving along the x-axis with a velocity of -3.00 m/s is 69 m/s.
To find the impulse of a force, which is equal to the change in momentum, we can use the equation:
Impulse = ∫ F dt
(a) To find the impulse of the force, we need to integrate the force function over time. From the given figure, let's assume that the scalar equation of the force is given by Fx = f(t). To find the impulse, we need to integrate this function over time:
Impulse = ∫ f(t) dt
You will need the specific equation describing the force as a function of time in order to proceed with the integration. Once you have that equation, you can evaluate the integral over the given time interval to find the impulse.
(b) To find the final velocity of the particle when it is initially at rest, we need to use the principle of impulse-momentum. The impulse is equal to the change in momentum:
Impulse = m * (vf - vi)
where m is the mass of the particle, vf is the final velocity, and vi is the initial velocity (which is 0 in this case). Rearranging the equation, we can solve for the final velocity:
vf = (Impulse / m) + vi
Substitute the value of the impulse from part (a) and the mass of the particle (given as 2.50 kg) to find the final velocity.
(c) To find the final velocity of the particle when it is initially moving along the x-axis with a velocity of -3.00 m/s, we can again use the principle of impulse-momentum. The impulse is equal to the change in momentum:
Impulse = m * (vf - vi)
where m is the mass of the particle, vf is the final velocity, and vi is the initial velocity (-3.00 m/s in this case). Rearranging the equation, we can solve for the final velocity:
vf = (Impulse / m) + vi
Substitute the value of the impulse from part (a), the mass of the particle, and the initial velocity to find the final velocity.
a. The impulse of a force is the area under the curve. The curve could be broken up into several geometric figures: a triangle, a rectangle, etc. The area of a triangle is base times height and the area of a rectangle is length times width.Use this rule for findinf I
b. I = p(f) - p(i) = mv(f) - mv(i) = mv(f),
(as v(i) = 0)
c. I = p(f) - p(i) = mv(f) - mv(i)