Question

A car of mass \(1000 \mathrm{~kg}\) travelling at \(32 \mathrm{~m} / \mathrm{s}\) clashes into a rear of a truck of mass \(8000 \mathrm{~kg}\) moving in the same direction with a velocity of \(4 \mathrm{~m} / \mathrm{s}\). After the collision the car bounces with a velocity of \(8 \mathrm{~ms}^{-1}\). The velocity of truck after the impact is \(\mathrm{m} / \mathrm{s}\) (B) 4 (C) 6 (D) 9 (A) 8

Short answer

The final velocity of the truck after the collision is 7 m/s.

Step-by-step solution

Write the conservation of linear momentum equation

We will use the law of conservation of linear momentum to solve this problem. The linear momentum before the collision must be equal to the linear momentum after the collision, which can be written as: \( m_1v_1+m_2v_2=(m_1+m_2)V_f \), where \(m_1\) and \(m_2\) are the masses of the car and the truck, respectively, \(v_1\) and \(v_2\) are their initial velocities, respectively, and \(V_f\) is the final velocity of the system after the collision.

Use the given values to calculate the linear momentum before the collision

We are given the masses and initial velocities of the car and the truck. Plug in these values into the equation above: \( (1000 \mathrm{kg})(32 \mathrm{m/s})+(8000 \mathrm{kg})(4 \mathrm{m/s})=(1000 \mathrm{kg}+8000 \mathrm{kg})V_f \).

Calculate the final velocity of the system after the collision

We know the velocity of the car after the collision, which is 8 m/s. Therefore, we can write the linear momentum after the collision as: \( (1000 \mathrm{kg})(8 \mathrm{m/s})+(8000 \mathrm{kg})V_f=(1000 \mathrm{kg}+8000 \mathrm{kg})V_f \).

Solve for the velocity of the truck after the collision

Now we have an equation with only one unknown, the final velocity of the truck, \(V_f\). Solve for \(V_f\): \( (1000 \mathrm{kg})(8 \mathrm{m/s})+(8000 \mathrm{kg})V_f=(1000 \mathrm{kg})(32 \mathrm{m/s})+(8000 \mathrm{kg})(4 \mathrm{m/s}) \), \( (8000 \mathrm{kg})V_f=(1000 \mathrm{kg})(24 \mathrm{m/s})+(8000 \mathrm{kg})(4 \mathrm{m/s}) \), \( (8000 \mathrm{kg})V_f=(24000+32000) \mathrm{kg} \cdot \mathrm{m/s} \), \(VM: V_f=(56000/8000) \mathrm{m/s} \), \(V_f=7 \mathrm{m/s} \). The final velocity of the truck after the collision is 7 m/s, which is not one of the given options. There might be a mistake in the problem statement or the provided answer choices.