A 5000 kg delivery van moving at 10 m/s collides with a 2000 kg car moving at −15 m/s. If the car's final…

Question

A 5000 kg delivery van moving at 10 m/s collides with a 2000 kg car moving at −15 m/s. If the car's final velocity is 5 m/s, what is the van's final velocity?

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the collision problem:

3.1:

Define a positive direction and list the given values.
Let "towards the right" be the positive direction.
Mass of delivery van ( $m_v$ ) = $5000 \text{ kg}$
Initial velocity of delivery van ( $v_{vi}$ ) = $+10 \text{ m/s}$
Mass of car ( $m_c$ ) = $2000 \text{ kg}$
Initial velocity of car ( $v_{ci}$ ) = $-15 \text{ m/s}$ (moving left)
Final velocity of car ( $v_{cf}$ ) = $+5 \text{ m/s}$ (moving right)
Apply the law of conservation of linear momentum.
The total momentum before the collision equals the total momentum after the collision:
$m_v v_{vi} + m_c v_{ci} = m_v v_{vf} + m_c v_{cf}$
Substitute the values into the equation and solve for $v_{vf}$ .
$(5000 \text{ kg})(+10 \text{ m/s}) + (2000 \text{ kg})(-15 \text{ m/s}) = (5000 \text{ kg})v_{vf} + (2000 \text{ kg})(+5 \text{ m/s})$ $50000 \text{ kg} \cdot \text{m/s} - 30000 \text{ kg} \cdot \text{m/s} = (5000 \text{ kg})v_{vf} + 10000 \text{ kg} \cdot \text{m/s}$ $20000 \text{ kg} \cdot \text{m/s} = (5000 \text{ kg})v_{vf} + 10000 \text{ kg} \cdot \text{m/s}$ $20000 \text{ kg} \cdot \text{m/s} - 10000 \text{ kg} \cdot \text{m/s} = (5000 \text{ kg})v_{vf}$ $10000 \text{ kg} \cdot \text{m/s} = (5000 \text{ kg})v_{vf}$ $v_{vf} = \frac{10000 \text{ kg} \cdot \text{m/s}}{5000 \text{ kg}}$ $v_{vf} = +2 \text{ m/s}$
The magnitude of the velocity of the delivery van immediately after the collision is $\boxed{\text{2 m/s}}$ .

3.2:
Calculate the total initial kinetic energy of the system.
$KE_{initial} = \frac{1}{2}m_v v_{vi}^2 + \frac{1}{2}m_c v_{ci}^2$ $KE_{initial} = \frac{1}{2}(5000 \text{ kg})(10 \text{ m/s})^2 + \frac{1}{2}(2000 \text{ kg})(-15 \text{ m/s})^2$ $KE_{initial} = \frac{1}{2}(5000)(100) + \frac{1}{2}(2000)(225)$ $KE_{initial} = 250000 \text{ J} + 225000 \text{ J}$ $KE_{initial} = 475000 \text{ J}$
Calculate the total final kinetic energy of the system.
Using $v_{vf} = +2 \text{ m/s}$ from 3.1 and $v_{cf} = +5 \text{ m/s}$ .
$KE_{final} = \frac{1}{2}m_v v_{vf}^2 + \frac{1}{2}m_c v_{cf}^2$ $KE_{final} = \frac{1}{2}(5000 \text{ kg})(2 \text{ m/s})^2 + \frac{1}{2}(2000 \text{ kg})(5 \text{ m/s})^2$ $KE_{final} = \frac{1}{2}(5000)(4) + \frac{1}{2}(2000)(25)$ $KE_{final} = 10000 \text{ J} + 25000 \text{ J}$ $KE_{final} = 35000 \text{ J}$
Calculate the change in kinetic energy.
$\Delta KE = KE_{final} - KE_{initial}$ $\Delta KE = 35000 \text{ J} - 475000 \text{ J}$ $\Delta KE = -440000 \text{ J}$
The change in the kinetic energy of the system is $\boxed{\text{-440000 J}}$ .

3.3:
Since the kinetic energy of the system is not conserved (it changed by $-440000 \text{ J}$ ), the collision is INELASTIC.

3.4:
Identify the given values and the impulse-momentum theorem for the car.
Mass of car ( $m_c$ ) = $2000 \text{ kg}$
Initial velocity of car ( $v_{ci}$ ) = $-15 \text{ m/s}$
Final velocity of car ( $v_{cf}$ ) = $+5 \text{ m/s}$
Collision duration ( $\Delta t$ ) = $0.4 \text{ s}$
The impulse-momentum theorem states that $F_{avg} \Delta t = \Delta p_c = m_c v_{cf} - m_c v_{ci}$ .
Substitute the values into the formula and solve for the average force ( $F_{avg}$ ).
$F_{avg} (0.4 \text{ s}) = (2000 \text{ kg})(+5 \text{ m/s}) - (2000 \text{ kg})(-15 \text{ m/s})$ $F_{avg} (0.4 \text{ s}) = 10000 \text{ kg} \cdot \text{m/s} - (-30000 \text{ kg} \cdot \text{m/s})$ $F_{avg} (0.4 \text{ s}) = 10000 \text{ kg} \cdot \text{m/s} + 30000 \text{ kg} \cdot \text{m/s}$ $F_{avg} (0.4 \text{ s}) = 40000 \text{ kg} \cdot \text{m/s}$ $F_{avg} = \frac{40000 \text{ kg} \cdot \text{m/s}}{0.4 \text{ s}}$ $F_{avg} = 100000 \text{ N}$
The magnitude of the force that the delivery van exerts on the car is $\boxed{\text{100000 N}}$ .

A 5000 kg delivery van moving at 10 m/s collides with a 2000 kg car moving at -15 m/s. If the car's final velocity is 5 m/s, what is the van's final velocity?

Answer

Step-by-step solution

Define a positive direction and list the given values.

Apply the law of conservation of linear momentum.

Substitute the values into the equation and solve for $v_{vf}$ .

Calculate the total initial kinetic energy of the system.

Calculate the total final kinetic energy of the system.

Calculate the change in kinetic energy.

Identify the given values and the impulse-momentum theorem for the car.

Substitute the values into the formula and solve for the average force ( $F_{avg}$ ).

Try one yourself

Still stuck on this one?

More Physics Questions