Model rocket engines burn for only a short time before using up all their propellant. Suppose a model rocket is launched from its stand at an angle \(60^{\circ}\) above the horizontal. The mass of the rocket is \(0.25 \mathrm{~kg}\) and its final speed is \(50 \mathrm{~m} / \mathrm{s}\). If the engine burns for \(1.25 \mathrm{~s}\), the impulse it gives to the rocket is most nearly (A) \(15.6 \mathrm{~N} \mathrm{~s}\) (B) \(12.5 \mathrm{~N} \mathrm{~s}\) (C) \(6.25 \mathrm{~N} \mathrm{~s}\) (D) \(\frac{\sqrt{3}}{2} \times 12.5 \mathrm{~N} \mathrm{~s}\) (E) \(12.5 \mathrm{~N}\)

In physics, momentum is a measure of the motion of an object and is a product of its mass and velocity. It helps to understand how an object will react when it interacts with other objects or forces. The momentum of an object can be broken down into horizontal and vertical components, especially when considering objects moving at an angle, like a launched rocket.

Momentum is a vector quantity, meaning it has both magnitude and direction, and it's conserved in isolated systems. The final momentum of our model rocket is calculated separately for horizontal and vertical components because the launch angle results in different motion along these directions.

The formula for momentum is: \[ P = mv \]where:

• \( P \) is the momentum,
• \( m \) is the mass of the object, and
• \( v \) is the velocity of the object.

In the exercise provided, we have a rocket of mass \(0.25\, \text{kg}\) and a final speed of \(50\, \text{m/s}\). By using trigonometric functions to separate the speed into horizontal and vertical components and multiplying by the mass, we obtain the final momentum in each direction.

Launch angle is a critical factor in determining the trajectory and distance traveled by any projectile, including our model rocket. It is the angle at which an object is propelled above the horizontal. The launch angle directly influences the horizontal and vertical components of an object's velocity.

When an object is launched at an angle, its initial velocity can be broken down into two components:

• Horizontal component \(V_x = v\text{cos}\theta\)
• Vertical component \(V_y = v\text{sin}\theta\)

The trigonometric functions sine and cosine are used here to resolve the velocity according to the 60° launch angle given in the example. These components are essential in calculating momentum and other projectile motion parameters like range, maximum height, and time of flight. The chosen angle impacts both the distance the rocket travels and its maximum height.

Average force is the constant amount of force that, if applied over a certain time interval, would produce the same change in momentum as actually happened over that time interval. In our model rocket setting, the force applied by the engine's thrust acts over the short burn time to change the rocket's momentum from zero (at rest) to a specific value (once the fuel has been spent).

The average force can be determined using the formula:\[ F_{avg} = \frac{\text{Change in momentum}}{\text{Time interval}} \]The impulse of a force is a related concept. It measures the effect of a force acting over time, which changes the momentum of an object. Impulse, represented as \(I\), is the product of the average force and the time duration. It can be expressed as:\[ I = F_{avg} \times\text{Time interval} \]In the solution for the exercise, impulse was specifically calculated for the vertical component of the average force during the engine burn, producing the rocket's change in vertical momentum.