A cannon on a flat railroad car travels to the east with its barrel tilted 30° above horizontal. It fires a cannonball at 50 m/s. At t = 0 s , the car, starting from rest, begins to accelerate to the east at 2.0 m/s². At what time should the cannon be fired to hit a target on the tracks that is 400 m to the east of the car's initial position? Assume that the cannonball is fired from ground level.

Verified step by step guidance

1. Step 1: Determine the horizontal component of the cannonball's velocity. Use the formula $v_{x} = v imes ext{cos}( heta)$, where $v = 50$ m/s and $ heta = 30^ ext{o}$.
2. Step 2: Calculate the time it takes for the cannonball to reach the target horizontally. Use the formula $x = v_{x} imes t$, where $x = 400$ m. Solve for $t$.
3. Step 3: Calculate the relative velocity of the cannonball with respect to the accelerating train. Since the train is accelerating, use the formula $v_{ ext{relative}} = v_{x} - a imes t$, where $a = 2.0$ m/s².
4. Step 4: Adjust the time calculation to account for the relative motion between the cannonball and the train. Use the formula $x = v_{ ext{relative}} imes t$ and solve for $t$ again with the new relative velocity.
5. Step 5: Verify the vertical motion of the cannonball to ensure it does not hit the ground before reaching the target. Use the formula $y = v_{y} imes t - 0.5 imes g imes t^2$, where $v_{y} = v imes ext{sin}( heta)$ and $g = 9.8$ m/s², and check if $y ext{ remains } ext{non-negative throughout the journey.}$

Key Concepts

Projectile Motion

Projectile motion refers to the motion of an object that is launched into the air and is subject to gravitational force. It can be analyzed in two dimensions: horizontal and vertical. The horizontal motion is uniform, while the vertical motion is influenced by gravity, leading to a parabolic trajectory. Understanding the components of the initial velocity and the effects of gravity is crucial for predicting the projectile's path.

Relative Motion

Relative motion involves analyzing the movement of an object in relation to another object. In this scenario, the cannon is on a moving railroad car, which introduces the need to consider both the cannonball's motion and the car's acceleration. The effective velocity of the cannonball must be calculated relative to the target's position, taking into account the car's increasing speed as it accelerates.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, these equations will be essential for determining the time it takes for the cannonball to reach the target, as well as for calculating the distance traveled by the railroad car during that time. Using these equations allows for solving for unknown variables in motion problems.