Projectile Motion Problem: Calculating Velocity, Distance, and Potential Energy

[Flanery]

Homework Statement

"A 45.0 kg projectile is fired from a gun. The elevation angle of the gun is 30.0°. the projectile is in the air for 48.0 s before it hits the deck at the same height as it was fired. Ignoring friction find: the projectile's initial velocity, the horizontal distance the projectile traveled, and the potential energy of the projectile at it's highest point in its trajectory.

Weight: 45 kg
Angle of launch: 30°
Total time in air: 48 s

Homework Equations

$$D = VT$$

The Attempt at a Solution

We haven't done projectile motion problems since the beginning of the year. It seems this problem is a random one on the homework assignments. I've completely forgotten how to calculate these. I drew out the motion of it but it's not ringing any bells.

 

[cepheid]

Welcome to PF!

By symmetry, you know that the time it takes to reach the max height is half of the total travel time (think about it if you're not sure).

Since you know the vertical acceleration, and the time taken to reach the max height (i.e. the time needed to reduce the vertical velocity to zero), you can determine the initial vertical velocity.

Since you know the angle of launch, you can use the above result to determine the initial horizontal velocity, and from that, the total initial velocity.

Once you know the horizontal speed, you can figure out the total horizontal distance travelled.

The max height comes from basic kinematics. You already know the time taken to reach it, and the acceleration.

I hope this helps.

 

[Flanery]

Since you know the vertical acceleration, and the time taken to reach the max height (i.e. the time needed to reduce the vertical velocity to zero), you can determine the initial vertical velocity.

Hey, thanks for your response! I don't quite understand how I have the vertical acceleration. Do I need to find it from the weight/gravity/time?

 

[cepheid]

Hey, thanks for your response! I don't quite understand how I have the vertical acceleration. Do I need to find it from the weight/gravity/time?

What defines projectile motion is that the particle is solely under the influence of gravity. That's the only force that acts on it. So, the vertical acceleration is due to gravity.

 

[rwisz]

As a scientist, it is important to approach problems with a systematic and logical approach. Let's break down this problem into smaller parts and use the given information to solve for the unknowns.

First, let's calculate the initial velocity of the projectile. We can use the equation Vx = Vcosθ, where Vx is the horizontal component of velocity and θ is the elevation angle. Plugging in the values, we get Vx = Vcos30°. Similarly, we can use the equation Vy = Vsinθ to calculate the vertical component of velocity. Plugging in the values, we get Vy = Vsin30°. Since the projectile is in the air for 48 seconds, we can use the equation D = VT to calculate the total distance traveled. Plugging in the values, we get D = (Vcos30°)(48s). This gives us the horizontal distance traveled by the projectile.

Next, we can use the equation Vf = Vi + at to calculate the final velocity of the projectile when it hits the deck. Since the projectile is at the same height as it was fired, the final vertical velocity is 0. We can use the equation Vf^2 = Vi^2 + 2ad to calculate the initial vertical velocity. Plugging in the values, we get 0 = (Vsin30°)^2 + 2(-9.8m/s^2)d. Solving for d, we get d = (Vsin30°)^2/19.6. This gives us the vertical distance traveled by the projectile.

Finally, we can use the equation PE = mgh to calculate the potential energy of the projectile at its highest point. Since the projectile is at the same height as it was fired, the height (h) is 0. Therefore, the potential energy at its highest point is also 0.

In summary, the initial velocity of the projectile is V = Vcos30°, the horizontal distance traveled is D = (Vcos30°)(48s), and the potential energy at its highest point is PE = 0. It is important to review the equations and principles of projectile motion to fully understand and solve this problem.