Question

The velocity of an object becomes double then its kinetic energy will be :

• A. Kinetic Energy does not depend on velocity
• B. Two times
• C. Four times
• D. Eight times

Hint:

The relationship \(\text{KE} = \frac{1}{2}mv^2\) shows that kinetic energy is proportional to the square of the velocity (\(\text{KE} \propto v^2\)). This means: - If velocity \(v\) is multiplied by a factor \(N\), the kinetic energy KE is multiplied by a factor \(N^2\). - If \(v\) doubles (factor of 2), KE increases by \(2^2 = 4\) times. - If \(v\) triples (factor of 3), KE increases by \(3^2 = 9\) times. This quadratic relationship has significant implications, for example, in vehicle safety.

Answer 1

The Correct Option is C

Concept: Kinetic energy (KE) is the energy an object possesses due to its motion. It depends on the object's mass and its velocity. Step 1: Recall the Formula for Kinetic Energy The kinetic energy of an object is given by the formula: \[ \text{KE} = \frac{1}{2} m v^2 \] where:
\(m\) is the mass of the object (assumed to be constant in this problem).
\(v\) is the velocity (or speed, since KE is a scalar) of the object. Step 2: Define Initial and Final States Let the initial velocity of the object be \(v_1\). The initial kinetic energy (\(\text{KE}_1\)) is: \[ \text{KE}_1 = \frac{1}{2} m v_1^2 \] The problem states that the velocity of the object becomes double. So, the new (final) velocity, let's call it \(v_2\), is: \[ v_2 = 2 v_1 \] Step 3: Calculate the New Kinetic Energy Now, we calculate the new kinetic energy (\(\text{KE}_2\)) using the new velocity \(v_2\): \[ \text{KE}_2 = \frac{1}{2} m v_2^2 \] Substitute the expression for \(v_2\) (\(2v_1\)) into this equation: \[ \text{KE}_2 = \frac{1}{2} m (2v_1)^2 \] Step 4: Simplify the Expression for New Kinetic Energy When we square \((2v_1)\), both the \(2\) and \(v_1\) are squared: \[ (2v_1)^2 = 2^2 \times v_1^2 = 4 v_1^2 \] Now substitute this back into the equation for \(\text{KE}_2\): \[ \text{KE}_2 = \frac{1}{2} m (4v_1^2) \] We can rearrange the terms to make the comparison clearer: \[ \text{KE}_2 = 4 \times \left(\frac{1}{2} m v_1^2\right) \] Step 5: Compare the New Kinetic Energy with the Initial Kinetic Energy Notice that the term in the parentheses, \(\left(\frac{1}{2} m v_1^2\right)\), is exactly the expression for the initial kinetic energy, \(\text{KE}_1\). So, we can write: \[ \text{KE}_2 = 4 \times \text{KE}_1 \] This shows that the new kinetic energy is four times the initial kinetic energy. Therefore, if the velocity of an object becomes double, its kinetic energy will be four times.