The graph below shows the force applied to a 4.0 kg cart initially at rest but free to move on a horizontal track. Forces (in N) are measured on the ordinate and times (in s) are measured on the abscissa. Use the following values for the graph: a = 4.9 and x = 2.0. Assume the graph consists of regularly spaced grid squares. Calculate the final velocity of the cart, after being subjected to the forces illustrated in the graph.
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To calculate the final velocity of the cart, we need to determine the area under the graph of the force vs. time. The area under the graph represents the impulse applied to the cart, which is equal to the change in momentum. By using Newton's second law of motion, we can relate the change in momentum to the final velocity of the cart.
First, let's identify the geometric shape(s) formed by the graph. Looking at the graph, we can see that the area under the graph consists of two parts: a rectangle and a triangle.
The rectangle can be formed by extending the horizontal line from the end of the force values to the x-axis. The rectangle's base is the value of x, which is given as 2.0 s. The height of the rectangle is the force at that point, which is given as 4.9 N. Therefore, the area of the rectangle is calculated as follows:
Area of rectangle = base x height
= 2.0 s x 4.9 N
= 9.8 Ns
Next, the triangle can be formed by drawing a line diagonally from the original point on the graph to the end of the force values. The base of the triangle is also equal to x, which is given as 2.0 s. The height of the triangle is the difference between the final force value and the starting force value. The starting force value is 0 N since the cart is initially at rest, and the final force value is 4.9 N. Therefore, the height of the triangle is calculated as follows:
Height of triangle = final force - starting force
= 4.9 N - 0 N
= 4.9 N
Now, calculate the area of the triangle:
Area of triangle = 0.5 x base x height
= 0.5 x 2.0 s x 4.9 N
= 4.9 Ns
Finally, to determine the total impulse (change in momentum), we add up the areas of the rectangle and triangle:
Total impulse = Area of rectangle + Area of triangle
= 9.8 Ns + 4.9 Ns
= 14.7 Ns
Since impulse is equal to the change in momentum, we can use the equation:
Impulse = change in momentum
= mass x change in velocity
The mass of the cart is given as 4.0 kg. Rearranging the equation, we can solve for the change in velocity:
Change in velocity = Impulse / mass
= 14.7 Ns / 4.0 kg
= 3.675 m/s
Therefore, the final velocity of the cart after being subjected to the forces illustrated in the graph is 3.675 m/s.