Problem 1 - Applied Mathematics

Question

A force of \(10 \mathrm{~N}\) acts on a mass of \(2 \mathrm{~kg}\) for three seconds. If the initial velocity was \(50 \mathrm{~ms}^{-1}\) what is the final velocity?

Step-by-Step Solution

Answer

The final velocity is 65 m/s.

Step 1: Identify the known values

The force applied is 10 N, the mass of the object is 2 kg, the time duration is 3 seconds, and the initial velocity is 50 m/s.

Step 2: Use Newton's Second Law

According to Newton’s Second Law, the acceleration can be calculated using the formula \(a = \frac{F}{m}\). Substitute the given values: \(a = \frac{10 \mathrm{~N}}{2 \mathrm{~kg}} = 5 \mathrm{~ms}^{-2}\).

Step 3: Calculate the change in velocity

The change in velocity (∆v) can be found using the formula \(\Delta v = a \times t\). Substitute the values for acceleration and time: \(\Delta v = 5 \mathrm{~ms}^{-2} \times 3 \mathrm{~s} = 15 \mathrm{~ms}^{-1}\).

Step 4: Find the final velocity

The final velocity (v_f) is the initial velocity (v_i) plus the change in velocity (∆v). Substitute the known values into the formula: \(v_f = v_i + \Delta v\). So, \(v_f = 50 \mathrm{~ms}^{-1} + 15 \mathrm{~ms}^{-1} = 65 \mathrm{~ms}^{-1}\).

Key Concepts

Force and Mass

Newton's Second Law of Motion states that the force applied to an object is directly proportional to the acceleration of that object and inversely proportional to its mass. This is described by the formula: \[ F = ma \] Here, \'F\' is the force applied, \'m\' is the mass of the object, and \'a\' is the acceleration. When you apply a force of 10 N to a mass of 2 kg, you can calculate the acceleration easily.
You just need to use the formula: \[ a = \frac{F}{m} \] Substituting the given values, we get: \[ a = \frac{10 \: \textrm{N}}{2 \: \textrm{kg}} = 5 \: \textrm{ms}^{-2} \] This means the object will accelerate at a rate of 5 meters per second squared when a force of 10 N is applied to a mass of 2 kg.

Acceleration Calculation

Acceleration is a change in velocity over time. Once we know the acceleration, we can use it to find out how the object’s velocity changes over a specified time period. In the given problem, we have an acceleration of 5 \textrm{ms}^{-2} acting for a time duration of 3 seconds.
To calculate the change in velocity, we use the formula: \[ \Delta v = a \times t \] Here, \'\Delta v\' is the change in velocity, \'a\' is the acceleration, and \'t\' is the time.
Substitute the known values: \[ \Delta v = 5 \: \textrm{ms}^{-2} \times 3 \: \textrm{s} = 15 \: \textrm{ms}^{-1} \] Therefore, the object’s velocity changes by 15 meters per second over the period of 3 seconds due to the applied force.

Change in Velocity

To find the final velocity of the object, we need to consider its initial velocity and the change in velocity due to the applied force. The final velocity \'v_f\' can be calculated by adding the initial velocity \'v_i\' to the change in velocity \'\Delta v\'. This can be expressed with the following formula: \[ v_f = v_i + \Delta v \] In the given problem, the initial velocity is 50 \textrm{ms}^{-1}, and the change in velocity is 15 \textrm{ms}^{-1}.
Substitute these values into the formula: \[ v_f = 50 \: \textrm{ms}^{-1} + 15 \: \textrm{ms}^{-1} = 65 \: \textrm{ms}^{-1} \] Therefore, the final velocity of the object is 65 meters per second after a force of 10 N is applied for 3 seconds.