Question

If the kinetic energy of a body moving with a velocity of $(2\vec{i}+3\vec{j}-4\vec{k})$ ms$^{-1}$ is 87 J, then the mass of the body is

• A. 3 kg
• B. 12 kg
• C. 9 kg
• D. 6 kg

Hint:

Kinetic energy is a scalar quantity that depends on the speed (magnitude of velocity), not the velocity vector itself. To find the kinetic energy from a velocity vector, you must first calculate the square of its magnitude: $v^2 = v_x^2 + v_y^2 + v_z^2$.

Answer 1

The Correct Option is D

Step 1: Write the formula for kinetic energy.
The kinetic energy (KE) of a body of mass $m$ moving with a speed $v$ is given by: \[ KE = \frac{1}{2}mv^2. \] Here, $v$ is the magnitude of the velocity vector $\vec{v}$.

Step 2: Calculate the speed of the body.
The velocity vector is given as $\vec{v} = 2\vec{i}+3\vec{j}-4\vec{k}$. The speed $v$ is the magnitude of this vector, $|\vec{v}|$. \[ v = |\vec{v}| = \sqrt{2^2 + 3^2 + (-4)^2} = \sqrt{4+9+16} = \sqrt{29} \text{ m/s}. \] Therefore, the square of the speed is $v^2 = 29$ (m/s)$^2$.

Step 3: Substitute the known values into the kinetic energy formula and solve for mass.
We are given KE = 87 J. \[ 87 = \frac{1}{2} m (29). \] Multiply both sides by 2: \[ 174 = 29m. \] Solve for $m$: \[ m = \frac{174}{29}. \] We can see that $29 \times 6 = (30-1) \times 6 = 180 - 6 = 174$. \[ \boxed{m = 6 \text{ kg}}. \]