Problem 36 - Physics for Scientists and Engineers with Modern Physics

Question

(II) You buy a plastic dart gun, and being a clever physics student you decide to do a quick calculation to find its maximum horizontal range. You shoot the gun straight up, and it takes 4.0 s for the dart to land back at the barrel. What is the maximum horizontal range of your gun?

Step-by-Step Solution

Answer

The maximum horizontal range of the gun is 9.8 meters.

Step 1: Analyze Vertical Motion

As the dart is shot straight up and takes a total of 4.0 s to go up and back down, we know the time to reach maximum height is half of that, so 2.0 s. Using the formula for time of flight in vertical motion, \[ t = \frac{2v_0 \sin \theta}{g} \] we substitute \( \theta = 90^\circ \) and \( g = 9.8 \, \text{m/s}^2 \) to find \( v_0 \sin \theta = g \cdot t/2 = 9.8 \cdot 2 = 9.8 \, \text{m/s} \).

Step 2: Calculate Initial Speed

Since the dart shot vertically upward fully goes up and returns in 4.0 s, the initial vertical velocity component \( v_0 \sin \theta = 9.8 \, \text{m/s} \). Hence, for vertical motion shot up, the initial speed \( v_0 = 9.8 \, \text{m/s} \).

Step 3: Maximum Range Equation

For horizontal range, the formula is \[ R = \frac{v_0^2 \sin 2\theta}{g} \]. For maximum range, we need the dart to shoot at \( \theta = 45^\circ \). Thus, \( \sin 2\theta = 1 \).

Step 4: Solve for Maximum Horizontal Range

Plug in the values into the range formula: \[ R = \frac{(9.8)^2}{9.8} = 9.8 \, \text{m} \] Therefore, the maximum horizontal range of the dart gun is 9.8 meters.

Key Concepts

Maximum Range

In projectile motion, the maximum range is the furthest distance a projectile can travel horizontally. This range is influenced by the initial speed of the projectile and the angle at which it is launched. To achieve the maximum range, the projectile must be launched at an ideal angle.

In the case of your dart gun, the ideal launch angle is 45 degrees. At this angle, the horizontal and vertical components of the initial velocity are equal, maximizing the distance the dart will cover. The range is calculated using the equation:

• \[ R = \frac{v_0^2 \sin 2\theta}{g} \]

where:- \( R \) is the range,- \( v_0 \) is the initial velocity,- \( \theta \) is the launch angle,- and \( g \) is the acceleration due to gravity, which is approximately 9.8 m/s².
To find the maximum range of your dart gun, it is important to compute its initial speed, and then use the equation with the ideal launch conditions.

Vertical Motion

Vertical motion is a key aspect of projectile motion. When a projectile is launched, it is under the influence of gravity, which affects its vertical motion. This component can be analyzed separately from the horizontal motion.

In the problem with the dart gun, the dart is shot vertically upwards. As it moves upward, gravity slows it down until it comes to a momentary stop at its peak height before returning back down. This symmetrical nature of motion means that the time taken to reach the maximum height is equal to the time taken to descend back to the start.

• The equation for the time of flight in vertical motion is \[ t = \frac{2 v_0 \sin \theta}{g} \]

Given that the dart takes a total of 4.0 seconds to go up and come back, the time to ascend to its peak is 2.0 seconds. Inserting these values helps compute the initial vertical velocity, enabling us to solve further for maximum range.

Initial Velocity

The initial velocity, often denoted as \( v_0 \), is the speed at which a projectile is launched. It is a crucial element for determining how far and how high the projectile will go.

• In this context, the initial velocity is necessary to calculate both the vertical motion in reaching maximum height and the horizontal distance or range achieved.
  
  

For the dart gun, the dart was fired straight up. To return in 4.0 seconds, the upward initial velocity component can be derived from:

• \[ v_0 \sin \theta = \frac{g \cdot t}{2} = 9.8 \cdot 2 = 9.8 \text{ m/s} \]

Here, since the dart was shot vertically, \( \theta = 90^\circ \) making the full speed equal to the vertical component. Understanding \( v_0 \) allows more precise calculations for optimizing the gun's launch angle to determine the maximum range.