Question

A car accelerates from rest to \( u \, \text{m/s} \). The energy spent in this process is \( E \, \text{J} \). The energy required to accelerate the car from \( u \, \text{m/s} \) to \( 2u \, \text{m/s} \) is \( nE \, \text{J} \). The value of \( n \) is.

Answer 1

Correct Answer: 3

The kinetic energy of an object is given by: \[ E_k = \frac{1}{2} m v^2, \] where \( m \) is the mass and \( v \) is the velocity. **Energy spent to accelerate from rest to \( u \):** \[ E_1 = \frac{1}{2} m u^2. \] **Energy spent to accelerate from \( u \) to \( 2u \):** The kinetic energy at \( 2u \) is: \[ E_2 = \frac{1}{2} m (2u)^2 = \frac{1}{2} m (4u^2) = 2m u^2. \] The kinetic energy at \( u \) is: \[ E_1 = \frac{1}{2} m u^2. \] The energy required to go from \( u \) to \( 2u \) is: \[ \Delta E = E_2 - E_1 = 2m u^2 - \frac{1}{2} m u^2. \] Simplify: \[ \Delta E = \frac{4}{2} m u^2 - \frac{1}{2} m u^2 = \frac{3}{2} m u^2. \] We know that \( E_1 = \frac{1}{2} m u^2 \), so: \[ \Delta E = 3E_1. \] Thus, \( n = 3 \), and the energy required to accelerate the car from \( u \, \text{m/s} \) to \( 2u \, \text{m/s} \) is \( \boxed{3E} \).