Problem 38 - College Physics

Question

A block of ice with a mass of 2.50 \(\mathrm{kg}\) is moving on a fric- tionless, horizontal surface. At \(t=0,\) the block is moving to the right with a velocity of magnitude 8.00 \(\mathrm{m} / \mathrm{s}\) . Calculate the magnitude and direction of the velocity of the block after each of the following forces has been applied for 5.00 s: (a) a force of 5.00 N directed to the right; (b) a force of 7.00 \(\mathrm{N}\) directed to the left.

Step-by-Step Solution

Answer

(a) 18.00 m/s to the right, (b) 6.00 m/s to the left.

Step 1: Understand the initial conditions

The block of ice has an initial velocity \(v_0 = 8.00\, \mathrm{m/s}\) to the right, and a mass \(m = 2.50\, \mathrm{kg}\). The initial velocity is positive, as it moves to the right.

Step 2: Calculate acceleration for part (a)

For the force of 5.00 \(\mathrm{N}\) directed to the right, use Newton's second law, \( F = ma \). Thus, \( a = \frac{F}{m} = \frac{5.00\, \mathrm{N}}{2.50\, \mathrm{kg}} = 2.00\, \mathrm{m/s^2} \).

Step 3: Determine the final velocity for part (a)

Using the equation for velocity \( v = v_0 + at \), substitute \( v_0 = 8.00\, \mathrm{m/s}, a = 2.00\, \mathrm{m/s^2}, \) and \( t = 5.00\, \mathrm{s} \): \( v = 8.00 + (2.00)(5.00) = 18.00\, \mathrm{m/s} \). The final velocity after the force is applied is 18.00 \(\mathrm{m/s}\) to the right.

Step 4: Calculate acceleration for part (b)

For the force of 7.00 \(\mathrm{N}\) directed to the left, use Newton's second law reverse direction, \( a = \frac{-7.00\, \mathrm{N}}{2.50\, \mathrm{kg}} = -2.80\, \mathrm{m/s^2} \).

Step 5: Determine the final velocity for part (b)

Using the equation for velocity \( v = v_0 + at \), substitute \( v_0 = 8.00\, \mathrm{m/s}, a = -2.80\, \mathrm{m/s^2}, \) and \( t = 5.00\, \mathrm{s} \): \( v = 8.00 + (-2.80)(5.00) = -6.00\, \mathrm{m/s} \). The negative sign indicates the direction is to the left.

Key Concepts

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the causes of this motion. When you're studying kinematics, you're looking at how objects move—something you could describe with graphs, equations, or words. A common approach is to determine velocity, acceleration, and displacement over a given period.

In the context of our exercise, kinematics helps describe how the ice block's velocity changes over time. The initial velocity, time of travel, and any forces acting upon it directly influence how these kinematic variables evolve. With forces acting, the block either speeds up or slows down depending upon the direction of force relative to its motion.

By knowing the initial conditions (initial velocity, direction, and forces), you can use kinematic equations to predict what happens to the object later in its journey.

Force and Motion

Force and motion are key concepts that explain why objects move and how they change their motion. According to Newton's Second Law, an object will only accelerate if there is a net force acting upon it. This law is often summed up as \( F = ma \), where \( F \) is force, \( m \) is mass, and \( a \) is acceleration.

• In the exercise, forces are applied to the ice block to alter its velocity.
• A 5 N force to the right results in acceleration in the same direction, while a 7 N force to the left changes the direction of acceleration.

Understanding how forces impact an object's motion can help predict outcomes like the ice block speeding up, slowing down, or changing direction. The resulting acceleration from these forces is crucial for determining changes in velocity.

Velocity Calculation

Velocity is more than just speed; it's speed with direction. Calculating velocity involves knowing the initial velocity, the time over which the force acts, and the acceleration resulting from that force.

To arrive at the final velocity, we use the formula \( v = v_0 + at \), where \( v_0 \) is the initial velocity, \( a \) is the acceleration, and \( t \) is the time period the force is applied.

• For part (a), the initial velocity is 8.00 m/s, the acceleration is 2.00 m/s², and the force is applied for 5.00 seconds. This gives a final velocity of 18.00 m/s to the right.
• However, for part (b), with a force to the left, the acceleration becomes -2.80 m/s², resulting in a final velocity of -6.00 m/s, indicating movement to the left.

So, by plugging these values into the velocity formula, you can determine exactly how the velocity changes in different scenarios.