Question

A $4.88$ -kg object with a speed of $31.4\mathrm{m}/\mathrm{s}$ strikes a steel plate at an angle of ${42.0}^{\circ }$ and rebounds at the same speed and angle (Fig. 6-19). What is the change (magnitude and direction) of the linear momentum of the object?

Short answer

The change in the linear momentum of the object is $306.624kg·m/s$ in the opposite direction of the initial momentum.

Step-by-step solution

Calculate Initial Momentum

Calculate the initial momentum of the object using the formula $\text{momentum}=mass\times velocity$. The initial momentum is $4.88kg\times 31.4m/s=153.312kg·m/s$ in the direction it was originally moving.

Calculate Final Momentum

Calculate the final momentum of the object after it rebounds. It's still $4.88kg\times 31.4m/s=153.312kg·m/s$, but the direction has changed i.e., it's still the same magnitude, but now in the opposite direction.

Calculate Change in Momentum

The change in momentum is the final momentum minus the initial momentum. Keeping in mind that the direction has changed, the change is $153.312kg·m/s-(-153.312kg·m/s)=306.624kg·m/s$.

Key concepts

Momentum Conservation

When we talk about momentum conservation, we're referring to a fundamental principle in physics that says that the total momentum of a closed system remains constant over time, as long as no external forces acting on it.
This means that if no external forces affect the system, the momentum before any event will be equal to the momentum after the event. This principle is extremely helpful in solving many real-world physics problems, like collisions or explosions.

• In the given exercise, the object has a certain momentum when it hits the steel plate.
• The object then rebounds while maintaining the same magnitude of momentum—meaning the total momentum before and after the impact is the same, although its direction changes.

Rebound Collision

A rebound collision is a common scenario where an object bounces back after hitting another object. This type of collision clearly involves a change in direction.
In our example, the object rebounds after hitting the steel plate. Although its speed remains the same, the direction is reversed.

• The physical interaction at the surface causes the object to change direction while perfectly maintaining its speed, which is crucial in understanding elastic collisions where kinetic energy and momentum are conserved.
• This change in direction is important because it leads to a significant change in the object's momentum, even though its speed does not change.

Velocity Calculation

To understand the exercise fully, it is important to know how to calculate velocity. Velocity is the speed of something in a given direction. In problems involving linear momentum, both the magnitude and direction of velocity play crucial roles.

• Initially, the object travels at a velocity of 31.4 m/s at a 42-degree angle towards the steel plate.
• After rebounding, its velocity remains 31.4 m/s, but the angle and hence the direction have changed.

The velocity change involved in a rebound collision is a key factor in determining the change in momentum. These factors result in a greater understanding of how direction and speed influence momentum calculations in physics. To summarize, understanding how to calculate and interpret both the magnitude and direction of velocity directly influences the evaluation of momentum changes in physical processes like collisions.