Rocket Launch Question Breakdown - workshop 1

Exam Question Practice: Rocket Launch Scenario

• The workshops focus on practicing exam-style questions.

• The module includes three sets of workshops, all centered around exam-type questions.

Scenario Overview

• Basic scenario: A rocket is launched from a certain distance (1.4 km) away from a British forces base.

• The key is to extract relevant information from the scenario.

• Launch angle: The missile is fired by hand at a launch angle of 27 degrees above the horizontal.

Key Data and Assumptions

• Rocket motor burn time: 3.7 seconds.

• During the rocket phase (while the motor is burning), the projectile travels in a straight line, overcoming gravity.

• Freefall Phase: After the motor cuts out, the projectile enters a freefall phase, following a parabolic trajectory.

• Important assumption: Air resistance is ignored for simplicity (Newtonian mechanics).

• Height at motor cut-out: 92 meters above the ground.

• Horizontal distance at motor cut-out: 181 meters.

• Velocity at motor cut-out: 110 meters per second (This becomes the initial velocity for the freefall phase, or 'U' in SUVAT).

Questions to be Addressed

1. What is the maximum height above launch level reached by the projectile?

2. How long will the projectile take to reach its maximum height?

3. What is the total flight time?

4. What will be the total range that the projectile will reach?

5. What will be the projectile's impact velocity and angle?

6. Does it hit the target?

General Approach

• Always draw a picture, especially for complex scenarios.

• Define the frame of reference (x and y axes).

Diagram and Labeling

• Draw x and y axes to define the frame of reference.

• Illustrate the rocket phase as a straight line.

• Indicate the point where the rocket motor cuts out.

• Draw the ground and label key values:

  • Launch angle ($\theta$)

  • Height at motor cut-out (92 meters).

  • Horizontal distance at motor cut-out (181 meters).

  • Rocket phase duration (3.7 seconds).

• Rocket Phase: Use the subscript 'r' to denote values specific to the rocket phase.

• Freefall Phase: Illustrate the parabolic trajectory after motor cut-out.

• Draw another set of axes at the point where freefall begins.

• Label the initial freefall angle as the same as the launch angle ($\theta$).

• Indicate the impact site.

Given Values

• $\theta$ = 27 degrees.

• u_f = 110 meters per second (initial freefall velocity).

Question 1: Maximum Height

• The goal is to calculate the total maximum height from the launch point.

• Calculate the additional height gained during the freefall phase and add it to the height at motor cut-out (92 meters).

• Focus on the y-direction distance.

X and Y Components

• Separate calculations into x and y components.

• Assumption: No drag or air resistance.

• Horizontal (x) direction: Constant velocity (velocity = distance / time).

• Vertical (y) direction: Acceleration due to gravity (SUVAT equations).

Y-Direction Data (Freefall)

• s_y = unknown (height gained during freefall)

• Initial Velocity: Resolve the initial freefall velocity (u_f) into its y-component.

  • uy = uf \sin(\theta)

  • u_y = 110 \sin(27) = 49.938 m/s

• Remember to use full values throughout to minimise rounding errors.

• At maximum height, the final velocity in the y direction (v_y) is zero.

• Acceleration due to gravity (a) = -9.8 m/s^2 (negative since it opposes upward motion).

SUVAT Equation

• Choose the appropriate SUVAT equation (without time) to solve for s_y.

  • v^2 = u^2 + 2as

  • s = \frac{v^2 - u^2}{2a}

• Plug in the values:

  • s_y = \frac{0^2 - (49.938)^2}{2 \times (-9.8)}

• s_y = 127.239 meters.

Total Height

• Add the height gained during freefall to the height at motor cut-out.

  • Total height = 127.239 + 92 = 219.239 meters

• Round the final answer appropriately (e.g., 219.24 meters).

Question 2: Time to Reach Maximum Height

• Calculate the time it takes to go from freefall start to maximum height.

• Use the same y-direction data as before.

• Choose an appropriate SUVAT equation involving time (v = u + at).

• Solve for t:

  • t = \frac{v - u}{a}

  • t_{up} = \frac{0 - 49.938}{-9.8} = 5.095 seconds.

Total Time to Maximum Height

• Add the time it took during the rocket phase (3.7 seconds) to the time in freefall (t_{up}).

  • Total time = 5.095 + 3.7 = 8.795 seconds (approximately 8.80 seconds).

Exam Tips

• Write clearly and methodically down the page.

• If you make a mistake, put a single line through it and continue neatly below.

• Show your working; 80% of the marks may be for the method.

• Include units in your answers.

Question 3: Total Flight Time

• Calculate the time it takes to go from maximum height back to the ground (y-direction).

• Define a new phase: downward motion.

• Consider redefining the positive y-direction as downward.

Downwards Data (Freefall)

• Distance in the y-direction (s_y) = 219.239 meters (total height).

• Initial velocity in the y-direction (u_y) = 0 m/s (starting from maximum height).

• Acceleration due to gravity (a) = 9.8 m/s^2 (positive since it's in the downward direction).

• Solve for t_{down} .

SUVAT Equation

• Use the SUVAT equation s = ut + (1/2)at^2. Since u = 0, this simplifies to s = (1/2)at^2.

• Rearrange to solve for t:

  • t = \sqrt{\frac{2s}{a}}

  • t_{down} = \sqrt{\frac{2 \times 219.239}{9.8}} = 6.689 seconds.

Total Time of Flight

• t{total} = t{up} + t{down} + t{rocket} (5.095+ 6.689 + 3.7 = 15.484)

• Add the time for upward freefall and downward freefall. And time in rocket flight.

  • t_{total} \\approx 15.48 seconds

Question 4: Total Range

• Total Range: the total horizontal distance missile can cover.

• Calculate the horizontal distance traveled during the freefall phase.

• Use the formula: velocity = distance / time (since there is no horizontal acceleration or deceleration).

  • Distance = velocity * time. We have to find horizontal distance and multiply it by time in the freefall phase.

Horizontal Calculation

• Initial velocity in the x-direction (ux) = uf \cos(\theta) . This means we resolve \u_f vector to its x component.

• Distance = velocity in the x-direction * total time in freefall (Tf). We multiply x component of the velocity to the overall time projectile spends int the air. Time in freefall means, it is equal to Tf = T{up} + T{down}

• Remember we are at constrant horizontal velocity, that means initial velocity is the final.

• sx = uf \cos(27) \times(t{up} + t{down})

• sx = 110 \cos(27) \times(5.095 + 6.689) \sx = 98.01 \times 11.784 = 1155.04. We know that during that time, missile travelled 1155 metres.

Total Range

• Total range = rocket horizontal distance + freefall horizontal distance.

• s_{total} = 181 + 1155.04 = 1336.04 to two decimal places

Final Question

• Will the projectile the target?. The distance to the target is at 1400 metres while the missile travelled at 1336 metres. It would fall short.

• The missile fall short at 63.96 metres.

Air Resistance Consideration

• Taking into consideration air resistance the missile would go further?

• Due to the drag forces, velocity is going to slow down as it goes.

• Physics outlines the absolute maximum distance when using this kind of formulas

Calculating impactVelocity and Angle

• The missile comes at an and horizontal angle and we have to resolve that. Velocity must be taken to come final velocity is on the y axis and missile can impact.

• We will bring the X and Y vectors together

• The triangles that can be formed at both sites are like the tiniest

• The workshops focus on practicing exam-style questions, enhancing understanding of the core concepts through practical application.

• The module is structured around three sets of workshops, each designed to tackle exam-type questions to build confidence and competence.

Scenario Overview

• Basic scenario: A rocket is launched from a distance of 1.4 km away from a British forces base, posing a challenge in real-world application of physics principles.

• The key is to meticulously extract relevant information from the scenario, which is crucial for accurate problem-solving.

• Launch angle: The missile is fired manually at an angle of 27 degrees above the horizontal, introducing vector components to the problem.

Key Data and Assumptions

• Rocket motor burn time: 3.7 seconds, indicating the duration of the thrust phase.

• During the rocket phase (while the motor is active), the projectile travels in a straight line, as thrust overcomes gravity.

• Freefall Phase: After the motor cuts out, the projectile enters a freefall phase, following a parabolic trajectory governed by gravity.

• Important assumption: Air resistance is ignored for simplicity, allowing the use of Newtonian mechanics without complex aerodynamic calculations.

• Height at motor cut-out: 92 meters above the ground, marking the transition from thrust to freefall.

• Horizontal distance at motor cut-out: 181 meters, indicating the horizontal displacement during powered flight.

• Velocity at motor cut-out: 110 meters per second; this becomes the initial velocity for the freefall phase and is denoted as 'U' in SUVAT equations.

Questions to be Addressed

1. What is the maximum height above launch level reached by the projectile? This tests the understanding of projectile motion.

2. How long will the projectile take to reach its maximum height? Assessing the ability to calculate time-dependent projectile parameters.

3. What is the total flight time? Evaluating comprehension of the entire trajectory duration.

4. What will be the total range that the projectile will reach? Measuring the capability to determine horizontal displacement.

5. What will be the projectile's impact velocity and angle? Requires vector component analysis at impact.

6. Does it hit the target? Determining if the projectile's range meets the specified target distance.

General Approach

• Always draw a picture to visualize the scenario, particularly for complex problems, to aid in understanding and setup.

• Define the frame of reference (x and y axes) to establish the coordinate system for calculations.

Diagram and Labeling

• Draw x and y axes to define the frame of reference, ensuring a clear coordinate basis.

• Illustrate the rocket phase as a straight line, representing the powered ascent.

• Indicate the point where the rocket motor cuts out, marking the start of freefall.

• Draw the ground and label key values:

  • Launch angle (\theta)

  • Height at motor cut-out (92 meters)

  • Horizontal distance at motor cut-out (181 meters)

  • Rocket phase duration (3.7 seconds)

• Rocket Phase: Use the subscript 'r' to denote values specific to the rocket phase, distinguishing them from freefall parameters.

• Freefall Phase: Illustrate the parabolic trajectory after motor cut-out to represent unpowered flight.

• Draw another set of axes at the point where freefall begins to reset the coordinate system for freefall analysis.

• Label the initial freefall angle as the same as the launch angle (\theta), assuming no abrupt change in direction at motor cut-out.

• Indicate the impact site to visualize the projectile's final landing point.

Given Values

• \theta = 27 degrees.

• u_f = 110 meters per second (initial freefall velocity).

Question 1: Maximum Height

• The goal is to compute the total maximum height the projectile attains from the launch point.

• Determine the additional height gained during the freefall phase and add it to the height at motor cut-out (92 meters).

• Focus on the y-direction distance, utilizing vertical motion principles.

X and Y Components

• Perform calculations separately for x and y components to dissect the problem.

• Assumption: Absence of drag or air resistance simplifies calculations by ensuring no horizontal deceleration.

• Horizontal (x) direction: Constant velocity (velocity = distance / time).

• Vertical (y) direction: Acceleration due to gravity (SUVAT equations).

Y-Direction Data (Freefall)

• s_y = unknown (height gained during freefall)

• Initial Velocity: Resolve the initial freefall velocity (u_f) into its y-component:

  • uy = uf \sin(\theta)

  • u_y = 110 \sin(27) = 49.938 m/s

• Ensure to use full values throughout calculations to minimize rounding errors.

• At maximum height, the final velocity in the y direction (v_y) is zero.

• Acceleration due to gravity (a) = -9.8 m/s^2 (negative since it opposes upward motion).

SUVAT Equation

• Select the SUVAT equation that does not involve time to directly solve for s_y:

  • v^2 = u^2 + 2as

  • s = \frac{v^2 - u^2}{2a}

• Substitute the known values:

  • s_y = \frac{0^2 - (49.938)^2}{2 \times (-9.8)}

• s_y = 127.239 meters.

Total Height

• Sum the height gained during freefall to the height at motor cut-out:

  • Total height = 127.239 + 92 = 219.239 meters

• Appropriately round the final answer, e.g., 219.24 meters.

Question 2: Time to Reach Maximum Height

• Determine the duration it takes for the projectile to ascend from the start of freefall to its maximum height.

• Employ the y-direction data from the previous question.

• Use a suitable SUVAT equation that includes time (v = u + at).

• Solve for t:

  • t = \frac{v - u}{a}

  • t_{up} = \frac{0 - 49.938}{-9.8} = 5.095 seconds.

Total Time to Maximum Height

• Sum the duration of the rocket phase (3.7 seconds) with the time in freefall (t_{up}):

  • Total time = 5.095 + 3.7 = 8.795 seconds (approximately 8.80 seconds).

Exam Tips

• Present your solutions clearly and methodically down the page for easy review.

• If a mistake is made, neatly strike through it with a single line and continue the correct solution below.

• Always show your working steps, as method marks can account for 80% of the total score.

• Ensure that units are explicitly included in your answers for completeness.

Question 3: Total Flight Time

• Calculate the duration for the projectile to descend from its maximum height back to the ground (y-direction).

• Define a new phase specifically for the downward motion.

• Consider redefining the positive y-direction as downward to simplify calculations.

Downwards Data (Freefall)

• Distance in the y-direction (s_y) = 219.239 meters (total height).

• Initial velocity in the y-direction (u_y) = 0 m/s (starting from maximum height).

• Acceleration due to gravity (a) = 9.8 m/s^2 (positive since it aligns with the downward direction).

• Solve for t_{down}.

SUVAT Equation

• Apply the SUVAT equation s = ut + (1/2)at^2. Given that u = 0, simplify to s = (1/2)at^2.

• Rearrange to find t:

  • t = \sqrt{\frac{2s}{a}}

  • t_{down} = \sqrt{\frac{2 \times 219.239}{9.8}} = 6.689 seconds.

Total Time of Flight

• t{total} = t{up} + t{down} + t{rocket} (5.095+ 6.689 + 3.7 = 15.484)

• Sum the time for upward freefall, downward freefall, and rocket flight.

• t_{total} \approx 15.48 seconds

Question 4: Total Range

• Total Range: Calculate the total horizontal distance covered by the missile.

• Compute the horizontal distance traveled during the freefall phase.

• Apply the formula: velocity = distance / time (no horizontal acceleration or deceleration).

  • Distance = velocity * time. Find horizontal distance and multiply it by time in the freefall phase.

Horizontal Calculation

• Initial velocity in the x-direction (ux) = uf \cos(\theta). Resolve u_f vector to its x component.

• Distance = velocity in the x-direction * total time in freefall (Tf). Multiply x component of the velocity to the overall time projectile spends in the air. Where time in freefall = Tf = T{up} + T{down}

• Constant horizontal velocity implies initial velocity equals final velocity.

• sx = uf \cos(27) \times(t{up} + t{down})

• sx = 110 \cos(27) \times(5.095 + 6.689) \sx = 98.01 \times 11.784 = 1155.04. During that time, missile travelled 1155 metres.

Total Range

• Total range = rocket horizontal distance + freefall horizontal distance.

• s_{total} = 181 + 1155.04 = 1336.04 (two decimal places)

Final Question

• Determine if the projectile hits the target. The target lies at 1400 meters, but the missile travels only 1336 meters. Therefore, it falls short.

• The missile falls short by 63.96 meters.

Air Resistance Consideration

• What happens when air resistance is considered?

• Drag forces will slow the projectile, affecting velocity.

• Physics laws define an absolute maximum distance without it.

Calculating impactVelocity and Angle

• The missile's final horizontal and vertical velocities must be resolved to determine impact velocity and angle.

• Combine X and Y vectors to get a resultant vector.

• Analyze the triangles at both impact sites.