3. The total work required to increase the speed of a certain car from rest to \( 2.0 \mathrm{~m} / \mathrm{s} \) is 1500 J . What is the mass of the car?

To find the mass of the car, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy. The kinetic energy (KE) of an object is given by the formula: \[ KE = \frac{1}{2} mv^2 \] where \( m \) is the mass of the object and \( v \) is its velocity. In this case, the car starts from rest (initial velocity \( u = 0 \)) and is accelerated to a final velocity of \( v = 2.0 \, \text{m/s} \). The work done on the car is given as \( W = 1500 \, \text{J} \). Since the car starts from rest, the change in kinetic energy is equal to the final kinetic energy: \[ W = KE = \frac{1}{2} mv^2 \] Substituting the values we have: \[ 1500 \, \text{J} = \frac{1}{2} m (2.0 \, \text{m/s})^2 \] Calculating \( (2.0 \, \text{m/s})^2 \): \[ (2.0 \, \text{m/s})^2 = 4.0 \, \text{m}^2/\text{s}^2 \] Now substituting this back into the equation: \[ 1500 \, \text{J} = \frac{1}{2} m (4.0 \, \text{m}^2/\text{s}^2) \] This simplifies to: \[ 1500 \, \text{J} = 2.0 m \] Now, solving for \( m \): \[ m = \frac{1500 \, \text{J}}{2.0} = 750 \, \text{kg} \] Thus, the mass of the car is \( \boxed{750 \, \text{kg}} \).