An object with cross section A is shot horizontally across frictionless ice. Its initial velocity is v₀ₓ at t₀ = 0 s. Air resistance is not negligible. Show that the velocity at time t is given by the expression $v_x=\frac{v_{0x}}{1+C_d\rho A{v}_{0x}t/2m}$.

Verified step by step guidance

1. Step 1: Begin by identifying the forces acting on the object. Since the ice is frictionless, the only force opposing the motion is air resistance. Air resistance is proportional to the square of the velocity, and its magnitude can be expressed as F_air = (1/2) * C𝓭 * p * A * vₓ², where C𝓭 is the drag coefficient, p is the air density, A is the cross-sectional area, and vₓ is the velocity at time t.
2. Step 2: Apply Newton's second law, F_net = m * a, where m is the mass of the object and a is its acceleration. The net force acting on the object is -F_air (negative because it opposes the motion). Thus, m * dvₓ/dt = -(1/2) * C𝓭 * p * A * vₓ².
3. Step 3: Rearrange the equation to isolate dvₓ/dt: dvₓ/dt = -(C𝓭 * p * A * vₓ²) / (2m). This is a first-order differential equation that relates the rate of change of velocity to the velocity itself.
4. Step 4: Solve the differential equation using separation of variables. Rewrite the equation as (dvₓ/vₓ²) = -(C𝓭 * p * A / (2m)) * dt. Integrate both sides: ∫(1/vₓ²) dvₓ = -∫(C𝓭 * p * A / (2m)) dt. The left-hand side integrates to -1/vₓ, and the right-hand side integrates to -(C𝓭 * p * A / (2m)) * t + constant.
5. Step 5: Solve for vₓ by isolating it. Use the initial condition vₓ = v₀ₓ at t = 0 to determine the constant of integration. After substituting the constant and rearranging, you obtain the expression vₓ = v₀ₓ / (1 + (C𝓭 * p * A * v₀ₓ * t) / (2m)).

Key Concepts

Drag Force

The drag force is a resistance force that acts opposite to the direction of motion of an object moving through a fluid, such as air. It is influenced by factors like the object's velocity, cross-sectional area, and the fluid's density. The drag force can be modeled using the equation F_d = 0.5 * C_d * ρ * A * v², where C_d is the drag coefficient, ρ is the fluid density, A is the cross-sectional area, and v is the velocity.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This relationship is expressed mathematically as F = m * a. In the context of the problem, the net force acting on the object is the drag force, which affects its horizontal velocity over time.

Exponential Decay in Velocity

In scenarios involving drag force, the velocity of an object can exhibit exponential decay over time. This occurs because the drag force, which is dependent on the velocity, reduces the object's speed as it moves. The derived expression for velocity over time reflects this decay, showing how the initial velocity is modified by the effects of drag, leading to a gradual decrease in speed as time progresses.