Question

A single force F(x) = −4.0x (in newtons) acts on a 1.0-kg body. When x = 3.5 m, the speed of the body is 4.0 m/s. What is its speed at x = 2.0 m?

a) 6.0 m/s
b) 8.0 m/s
c) 5.0 m/s
d) 7.0 m/s

Answer 1

To find the speed of the body at x = 2.0 m, we can use the principle of conservation of mechanical energy. At x = 3.5 m, the speed of the body is 4.0 m/s. Using the equation KE = 0.5mv^2, we can calculate the initial and final kinetic energies and solve for the final speed. The final speed of the body at x = 2.0 m is 4.0 m/s.

Step-by-step explanation:

To find the speed of the body at x = 2.0 m, we can use the principle of conservation of mechanical energy. At x = 3.5 m, the speed of the body is given as 4.0 m/s. We can calculate the initial potential energy of the body at x = 3.5 m using the equation U = mgh, where m is the mass of the body, g is the acceleration due to gravity, and h is the height. Since the body is moving horizontally, the height remains constant, and the change in potential energy is zero. Therefore, the initial kinetic energy at x = 3.5 m is equal to the mechanical energy of the body. Using the equation KE = 0.5mv^2, we can calculate the initial kinetic energy. Assuming no external forces acting on the body, the mechanical energy is conserved. At x = 2.0 m, the body's potential energy is given as U = mgh, and since the height remains constant, the change in potential energy is zero. Therefore, the final kinetic energy at x = 2.0 m is equal to the mechanical energy of the body. Using the equation KE = 0.5mv^2, we can solve for the final speed of the body at x = 2.0 m.

Given: F(x) = -4.0x N, m = 1.0 kg, x1 = 3.5 m, v1 = 4.0 m/s, x2 = 2.0 m

To find the final speed v2:

1. Calculate the initial kinetic energy using KE = 0.5mv1^2
2. Calculate the final kinetic energy using KE = 0.5mv2^2
3. Set the initial kinetic energy equal to the final kinetic energy and solve for v2

Plugging in the given values:

1. Initial kinetic energy: KE1 = 0.5 * 1.0 kg * (4.0 m/s)^2 = 8.0 J
2. Final kinetic energy: KE2 = 0.5 * 1.0 kg * v2^2
3. Setting the initial and final kinetic energies equal: KE1 = KE2 => 8.0 J = 0.5 * 1.0 kg * v2^2

Simplifying the equation:

1. 16 = v2^2
2. Taking the square root of both sides: v2 = ±4.0 m/s

Since speed cannot be negative, the final speed of the body at x = 2.0 m is 4.0 m/s. Therefore, the correct answer is d) 4.0 m/s.