Problem 6 - Applied Mathematics

Question

A body of mass \(m\) is moving with speed \(v\) when a constant force \(F\) newtons is applied to it in the direction of motion for \(t\) seconds: (a) The impulse of the force is \(F t\) newton second. (b) \(F t=m v\), (c) The body loses an amount of kinetic energy equal to \(F t\). (d) The final speed of the body is \(v+\frac{F t}{m}\).

Step-by-Step Solution

Answer

Statements (a), (b), and (d) are correct; statement (c) is incorrect.

Step 1: Understanding Impulse

Impulse is given by the product of the force and the time for which it is applied. Therefore, the impulse of the force is indeed: \[ I = F \times t \] This confirms statement (a) is correct.

Step 2: Relating Impulse to Momentum Change

Impulse can also be related to the change in momentum of the body. If a body of mass \( m \) is subjected to a force \( F \) for a time \( t \), the impulse \( F \times t \) equals the change in momentum: \[ F \times t = \text{change in momentum} \]

Step 3: Initial and Final Momentum

Since the force is applied in the direction of motion, the initial momentum of the body is \( m \times v \). The final momentum after time \( t \) is \( m \times v_{f} \) where \( v_f \) is the final speed.

Step 4: Calculating the Change in Momentum

The change in momentum is: \[ \text{change in momentum} = m \times v_{f} - m \times v \] From the impulse-momentum theorem we get: \[ F \times t = m \times v_{f} - m \times v \]

Step 5: Finding the Final Speed

Rearranging the equation: \[ F \times t = m \times (v_{f} - v) \] Therefore, \[ v_{f} = v + \frac{ F \times t }{ m } \] This confirms statement (d) is correct.

Step 6: Kinetic Energy Consideration

The initial kinetic energy is \( \frac{1}{2} m v^2 \). After the force is applied, the final kinetic energy is \( \frac{1}{2} m (v + \frac{ F \times t }{ m })^2 \). The change in kinetic energy is given by: \[ \Delta KE = \frac{1}{2} m (v + \frac{ F \times t }{ m })^2 - \frac{1}{2} m v^2 \] Simplifying this is not straightforward and does not simply equal \( F \times t \), so statement (c) is incorrect.

Key Concepts

Impulse

When we talk about 'Impulse' in physics, we're discussing the product of force and the time duration over which the force is applied. Mathematically, impulse is described as:
\[ I = F \times t \]. The impulse can be visualized as the push or pull experienced by an object over a certain period. For example, in our given exercise, a constant force is applied over a specific duration, and this generates an impulse. Impulse is significant because it relates directly to the change in momentum of an object. To sum up, impulse provides a handy way of looking at sudden forces acting over short times.

Momentum

Momentum, symbolized as \(p\), is the product of an object's mass \(m\) and its velocity \(v\). Essentially, momentum measures how hard it is to stop a moving object. If you consider our problem, the initial momentum of the body is represented by \(m \times v\). When a force acts on it for a time \(t\), it changes this momentum. This can be expressed using the impulse-momentum theorem, which states:
\[ F \times t = \text{change in momentum} \]. So, after time \(t\), the body's final momentum is \(m \times v_f\), where \(v_f\) is the new speed. By understanding momentum, it's easier to grasp how forces cause changes in the motion of an object.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is given by the equation:
\[ KE = \frac{1}{2} m v^2 \]. In the given exercise, the initial kinetic energy of the body before the force is applied is \(\frac{1}{2} m v^2\). After applying the force, the final kinetic energy becomes \(\frac{1}{2} m (v_f)^2\). The change in kinetic energy is more complex to compute directly, and it doesn't simply equate to the impulse (\(F \times t\)). This discrepancy shows kinetic energy's more intricate relationship with force and motion, emphasizing that energy transformations aren't always straightforward.

Force

Force is an interaction that changes the motion of an object. In our context, the constant force \(F\) is applied in the same direction as the object's motion. Using Newton's second law, we know force is related to mass and acceleration by \(F = m \times a\). When force is applied over time, it changes the object's momentum, calculated through impulse. Constant force means that the object continually accelerates while the force is applied. This results in a final velocity \(v_f\) which can be determined by the equation:
\[ v_f = v + \frac{F \times t}{m} \]. This clearly ties force to changes in motion, showing how sustained force affects an object's velocity and overall movement.