Kinematic Equations for Projectile Motion

Oct 13, 2025 by talynt.org 34 views

Understanding the Physics Behind the T-Shirt Cannon

Projectile Motion is a fundamental concept in physics that describes the motion of objects thrown or launched into the air. When a t-shirt cannon fires a shirt straight up, it follows a parabolic trajectory under the influence of gravity. To determine the maximum height reached by the t-shirt, we need to apply the principles of kinematics and dynamics.

Kinematic Equations for Projectile Motion

The kinematic equations for projectile motion are given by:

Horizontal motion: x = x0 + v0x * t
Vertical motion: y = y0 + v0y * t - (1/2) * g * t^2

where:

x and y are the horizontal and vertical positions of the object
x0 and y0 are the initial positions
v0x and v0y are the initial horizontal and vertical velocities
g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth)
t is time

Applying the Kinematic Equations to the T-Shirt Cannon

In this case, the t-shirt is fired straight up, so the initial horizontal velocity (v0x) is zero. The initial vertical velocity (v0y) is equal to the muzzle velocity of the cannon, which is 18 m/s. We can assume that the initial position (x0 and y0) is at the ground level, which is zero.

Calculating the Time of Flight

To find the maximum height reached by the t-shirt, we need to calculate the time of flight (t). We can use the vertical motion equation:

y = y0 + v0y * t - (1/2) * g * t^2

Since the t-shirt starts from the ground level (y0 = 0), the equation simplifies to:

y = v0y * t - (1/2) * g * t^2

We want to find the time when the t-shirt reaches its maximum height, which means the vertical velocity (v0y) is zero. We can set up the equation:

0 = v0y * t - (1/2) * g * t^2

Substituting v0y = 18 m/s and g = 9.8 m/s^2, we get:

0 = 18t - 4.9t^2

Rearranging the equation, we get:

4.9t^2 - 18t = 0

Factoring out t, we get:

t(4.9t - 18) = 0

This gives us two possible solutions:

t = 0 (initial time) t = 18/4.9 ≈ 3.67 s (time of flight)

Calculating the Maximum Height

Now that we have the time of flight, we can calculate the maximum height reached by the t-shirt. We can use the vertical motion equation:

y = v0y * t - (1/2) * g * t^2

Substituting v0y = 18 m/s, t = 3.67 s, and g = 9.8 m/s^2, we get:

y = 18 * 3.67 - (1/2) * 9.8 * (3.67)^2 y ≈ 66.06 - 66.06 y ≈ 0 m

However, this is not the maximum height reached by the t-shirt. The t-shirt will continue to rise until its vertical velocity becomes zero. To find the maximum height, we need to use the equation:

v = v0 - g * t

where v is the final vertical velocity (zero at the maximum height). We can set up the equation:

0 = 18 - 9.8 * t

Substituting t = 3.67 s, we get:

0 = 18 - 9.8 * 3.67 0 = 18 - 35.86 0 = -17.86

This is not correct, as the t-shirt will not reach a negative height. The error is due to the fact that we assumed the t-shirt reaches its maximum height at the time of flight (t = 3.67 s). In reality, the t-shirt will continue to rise after the time of flight.

Correcting the Calculation

To find the maximum height, we need to use the equation:

y = v0y * t - (1/2) * g * t^2

We can substitute v0y = 18 m/s and g = 9.8 m/s^2, but we need to find the correct time (t) when the t-shirt reaches its maximum height. We can use the equation:

v = v0 - g * t

where v is the final vertical velocity (zero at the maximum height). We can set up the equation:

0 = 18 - 9.8 * t

Rearranging the equation, we get:

t = 18/9.8 ≈ 1.84 s

This is the time when the t-shirt reaches its maximum height. We can now substitute this value of t into the equation:

y = v0y * t - (1/2) * g * t^2

Substituting v0y = 18 m/s, t = 1.84 s, and g = 9.8 m/s^2, we get:

y = 18 * 1.84 - (1/2) * 9.8 * (1.84)^2 y ≈ 33.12 - 16.51 y ≈ 16.61 m

Conclusion

The t-shirt cannon fires a shirt straight up at 18 m/s. To determine the maximum height reached by the t-shirt, we applied the principles of kinematics and dynamics. We calculated the time of flight and the maximum height reached by the t-shirt. The maximum height is approximately 16.61 meters.

Frequently Asked Questions

What is the time of flight for the t-shirt? The time of flight for the t-shirt is approximately 3.67 seconds.
What is the maximum height reached by the t-shirt? The maximum height reached by the t-shirt is approximately 16.61 meters.
What is the acceleration due to gravity? The acceleration due to gravity is approximately 9.8 m/s^2 on Earth.

References

Projectile Motion. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Projectile_motion
Kinematics. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Kinematics
Dynamics. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Dynamics_(physics)

Understanding the Physics Behind the T-Shirt Cannon

Q&A Session

Q: What is the time of flight for the t-shirt?

A: The time of flight for the t-shirt is approximately 3.67 seconds.

Q: What is the maximum height reached by the t-shirt?

A: The maximum height reached by the t-shirt is approximately 16.61 meters.

Q: What is the acceleration due to gravity?

A: The acceleration due to gravity is approximately 9.8 m/s^2 on Earth.

Q: How does the t-shirt cannon work?

A: The t-shirt cannon uses a combination of air pressure and mechanical energy to launch the t-shirt into the air. The cannon is typically powered by a compressed air tank or a spring-loaded mechanism.

Q: What are the factors that affect the trajectory of the t-shirt?

A: The factors that affect the trajectory of the t-shirt include the initial velocity, the angle of launch, the mass of the t-shirt, and the air resistance.

Q: Can the t-shirt cannon be used for other purposes?

A: Yes, the t-shirt cannon can be used for other purposes such as launching other objects, like balls or even small toys.

Q: How accurate is the t-shirt cannon?

A: The accuracy of the t-shirt cannon depends on various factors such as the design of the cannon, the quality of the materials used, and the skill of the operator.

Q: Can the t-shirt cannon be used in a competitive setting?

A: Yes, the t-shirt cannon can be used in a competitive setting such as a sporting event or a festival.

Q: What are the safety precautions that need to be taken when using the t-shirt cannon?

A: The safety precautions that need to be taken when using the t-shirt cannon include wearing protective gear, ensuring that the area is clear of obstacles, and following the manufacturer's instructions.

Additional Resources

Projectile Motion. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Projectile_motion
Kinematics. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Kinematics
Dynamics. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Dynamics_(physics)
T-Shirt Cannon. (n.d.). Retrieved from https://en.wikipedia.org/wiki/T-shirt_cannon

Conclusion

The t-shirt cannon is a fun and interesting device that can be used to launch t-shirts into the air. By understanding the physics behind the t-shirt cannon, we can determine the maximum height reached by the t-shirt and other factors that affect its trajectory. The t-shirt cannon can be used for various purposes, including competitive settings, and requires safety precautions to be taken when operating it.

Frequently Asked Questions

What is the time of flight for the t-shirt? The time of flight for the t-shirt is approximately 3.67 seconds.
What is the maximum height reached by the t-shirt? The maximum height reached by the t-shirt is approximately 16.61 meters.
What is the acceleration due to gravity? The acceleration due to gravity is approximately 9.8 m/s^2 on Earth.
How does the t-shirt cannon work? The t-shirt cannon uses a combination of air pressure and mechanical energy to launch the t-shirt into the air.
What are the factors that affect the trajectory of the t-shirt? The factors that affect the trajectory of the t-shirt include the initial velocity, the angle of launch, the mass of the t-shirt, and the air resistance.

References

Projectile Motion. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Projectile_motion
Kinematics. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Kinematics
Dynamics. (n.d.). Retrieved from https://en.wikipedia.org/wiki/Dynamics_(physics)
T-Shirt Cannon. (n.d.). Retrieved from https://en.wikipedia.org/wiki/T-shirt_cannon