Question

A 1050-kg sports car is moving westbound at 15.0 m/s on a level road when it collides with a 6320-kg truck driving east on the same road at 10.0 m/s. The two vehicles remain locked together after the collision. (a) What is the velocity (magnitude and direction) of the two vehicles just after the collision? (b) At what speed should the truck have been moving so that both it and the car are stopped in the collision? (c) Find the change in kinetic energy of the system of two vehicles for the situations of part (a) and part (b). For which situation is the change in kinetic energy greater in magnitude?

Short answer

(a) 6.44 m/s east; (b) 2.49 m/s; (c) Part (a): 418855 J, Part (b): 137680.44 J. Magnitude is greater in (a).

Step-by-step solution

Understand the Conservation of Momentum

The law of conservation of momentum states that in the absence of external forces, the total momentum of a system remains constant before and after a collision. The formula for momentum is given by \( p = mv \), where \( m \) is the mass and \( v \) is the velocity.

Calculate Initial Total Momentum

Calculate the initial momentum for both vehicles. The sports car has a momentum of \( p_{car} = 1050 \text{ kg} \times 15.0 \text{ m/s} = 15750 \text{ kg m/s west} \). The truck has a momentum of \( p_{truck} = 6320 \text{ kg} \times 10.0 \text{ m/s} = 63200 \text{ kg m/s east} \). Since they are moving in opposite directions, one of these will be negative: therefore the net initial momentum is \( 15750 - 63200 = -47450 \text{ kg m/s} \).

Using Conservation of Momentum to Find Final Velocity

After the collision, both vehicles move together as one mass \( (m_{total} = 1050 + 6320 = 7370 \text{ kg}) \). Since momentum is conserved: initial momentum = final momentum. Thus, \( -47450 = 7370 \times V \). Solving for \( V \), we have \( V = \frac{-47450}{7370} \approx -6.44 \text{ m/s} \). The velocity is 6.44 m/s, directed east.

Calculate Required Truck Speed for Complete Stop

For the total system to come to a stop after the collision, the initial momentum must be zero. Therefore, set the car's westbound momentum and the truck's eastbound momentum equal: \( 1050 \times 15 = 6320 \times v_{truck} \). Solving, we have \( v_{truck} = \frac{15750}{6320} \approx 2.49 \text{ m/s east} \).

Calculate Change in Kinetic Energy for Part (a)

Calculate initial kinetic energies: car \( KE_{car} = 0.5 \times 1050 \times 15.0^2 \), truck \( KE_{truck} = 0.5 \times 6320 \times 10.0^2 \). Total initial KE = \( 118125 + 316000 = 434125 \text{ J} \). Final kinetic energy of combined mass after collision \( KE_{final} = 0.5 \times 7370 \times 6.44^2 \approx 15270 \text{ J} \). Change in KE = \( 434125 - 15270 = 418855 \text{ J} \).

Calculate Change in Kinetic Energy for Part (b)

In this scenario, the truck needs to move at 2.49 m/s for the system to stop, so the final kinetic energy is 0. Calculate initial kinetic energy with \( v_{truck} = 2.49 \text{ m/s} \): truck \( KE_{truck} = 0.5 \times 6320 \times 2.49^2 \approx 19555.44 \text{ J} \). Total initial KE with revised truck speed = \( 118125 + 19555.44 = 137680.44 \text{ J} \). Change in KE = \( 137680.44 - 0 = 137680.44 \text{ J} \).

Compare Kinetic Energy Changes

The change in kinetic energy for scenario (a) is 418855 J and for scenario (b) is 137680.44 J. The change in kinetic energy is greater in magnitude for scenario (a).

Key concepts

Kinetic Energy

Kinetic Energy is a fundamental concept in physics that quantifies the energy possessed by an object due to its motion. It is expressed by the equation: \[KE = \frac{1}{2}mv^2\]where

• \(m\) stands for mass.
• \(v\) represents velocity.

To find the kinetic energy of any moving object, multiply half its mass by the square of its speed. In our example involving a sports car and a truck, the kinetic energies of both vehicles initially are calculated in this way. The car, weighing 1050 kg and moving at 15 m/s, has a kinetic energy of 118,125 Joules. Meanwhile, the heavier truck at 6320 kg moving at 10 m/s possesses 316,000 Joules. The total kinetic energy before they collide is the sum of these separate energies. Understanding how kinetic energy changes helps us recognize lost or transformed energy during collisions.

Inelastic Collision

An Inelastic Collision refers to a type of collision where the colliding entities stick together post-impact, losing some kinetic energy in the process. The perfect example from our exercise involves the car and truck when they lock together upon and after colliding.

• The defining characteristic of inelastic collisions is that energy is not conserved. This means kinetic energy before the collision is higher compared to afterwards.
• Even though kinetic energy is lost, momentum is still preserved.

In such cases, the two bodies behave as a single system with a combined mass, dramatically affecting their movements' energy distribution. The result is a noticeable decrease in the system's kinetic energy, as evidenced by the lower energy calculations for the combined mass post-collision.

Momentum Calculation

Momentum Calculation is essential to understand movement in physics, especially in collision scenarios. It involves assessing the product of mass and velocity of moving objects, capturing a snapshot of how momentum shifts through interactions.
The momentum equation, represented as:\[p = m \times v\]allows for such analyses. For the car and truck problem, calculating individual momenta revealed how their opposite directions contributed to collective momentum. It was found:

• The car's momentum was 15,750 kg·m/s directed west.
• The truck's momentum was 63,200 kg·m/s moving east.

Subtracting them due to opposite directions resulted in a net momentum of -47,450 kg·m/s, providing critical insight into their combined velocity post-collision. This evidence of momentum conservation allows us to compute the velocity of both locked vehicles according to their total mass.