A dockworker applies a constant horizontal force of $80.0$ N to a block of ice on a smooth horizontal floor. The frictional force is negligible. The block starts from rest and moves $11.0$ m in $5.00$ s. What is the mass of the block of ice?

Verified step by step guidance

1. Start by identifying the known quantities: the applied force \( F = 80.0 \, \text{N} \), the displacement \( d = 11.0 \; \text{m} \), the time \( t = 5.00 \; \text{s} \), and the fact that the block starts from rest (initial velocity \( v_0 = 0 \; \text{m/s} \)). The frictional force is negligible, so the net force is equal to the applied force.
2. Use the kinematic equation \( d = v_0 t + \frac{1}{2} a t^2 \) to solve for the acceleration \( a \). Since \( v_0 = 0 \), the equation simplifies to \( d = \frac{1}{2} a t^2 \). Rearrange to find \( a \): \( a = \frac{2d}{t^2} \). Substitute \( d = 11.0 \; \text{m} \) and \( t = 5.00 \; \text{s} \) into the equation.
3. Once the acceleration \( a \) is determined, use Newton's second law \( F = ma \) to solve for the mass \( m \). Rearrange the equation to find \( m \): \( m = \frac{F}{a} \). Substitute \( F = 80.0 \; \text{N} \) and the calculated value of \( a \) into this equation.
4. Perform the calculations step by step to determine the value of \( a \) and then \( m \). Ensure that the units are consistent throughout the calculations (e.g., \( \text{N} = \text{kg} \cdot \text{m/s}^2 \)).
5. Finally, verify the result by checking that the calculated mass \( m \) and acceleration \( a \) satisfy both the kinematic equation and Newton's second law. This ensures the solution is consistent and correct.

Key Concepts

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 by the formula F = ma, where F is the net force, m is the mass, and a is the acceleration. In this scenario, the constant horizontal force applied to the block of ice will cause it to accelerate, allowing us to determine its mass.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this case, we can use the equation s = ut + (1/2)at², where s is the displacement, u is the initial velocity, a is the acceleration, and t is the time, to find the acceleration of the block of ice and subsequently its mass.

Acceleration

Acceleration is defined as the rate of change of velocity of an object with respect to time. It can be calculated by dividing the change in velocity by the time taken for that change. In this problem, since the block starts from rest, we can determine its acceleration using the distance it travels and the time taken, which will then help us apply Newton's Second Law to find the mass.