Problem 42 - College Physics

Question

For a scene in a movie, a stunt driver drives a \(1.50 \times 10^{3} \mathrm{~kg} \mathrm{SUV}\) with a length of \(4.25 \mathrm{~m}\) around a circular curve with a radius of curvature of \(0.333 \mathrm{~km}\) (vFig. 7.34\()\). The vehicle is to be driven off the edge of a gully \(10.0 \mathrm{~m}\) wide, and land on the other side \(2.96 \mathrm{~m}\) below the initial side. What is the minimum centripetal acceleration the SUV must have in going around the circular curve to clear the gully and land on the other side?

Step-by-Step Solution

Answer

Minimum centripetal acceleration is approximately \(0.506 \text{ m/s}^2\).

Step 1: Understand the Problem

The goal is to find the minimum centripetal acceleration required for an SUV to successfully jump across a gully that is 10 meters wide and land 2.96 meters lower. The vehicle travels in a circular path before the jump.

Step 2: Convert Units

Convert the radius of curvature from kilometers to meters since other dimensions are given in meters. Thus, \(0.333 \text{ km} = 333 \text{ m}\).

Step 3: Calculate Time to Cross the Gully

To clear the gully, the horizontal distance covered is 10 m. Use the equations of projectile motion. Assuming horizontal velocity \(v_x\) and the drop \(h = 2.96 \text{ m}\):1. Time \( t \) to fall a height \( h \) is given by the equation: \[ h = \frac{1}{2} g t^2 \] Solving for \( t \) yields: \[ t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2 \times 2.96}{9.81}} \approx 0.77 \text{ s} \].

Step 4: Calculate Horizontal Velocity

Using the time \( t \) obtained, the horizontal velocity \( v_x \) needed to cross the gully is: \[ v_x \times t = 10 \] Solving for \( v_x \): \[ v_x = \frac{10}{0.77} \approx 12.99 \text{ m/s} \]

Step 5: Relate Horizontal Velocity to Centripetal Acceleration

Since the SUV is traveling in a circular path, its horizontal velocity is also the tangential velocity at the point of launch. The centripetal acceleration \( a_c \) is related to the tangential velocity and radius of curvature by: \[ a_c = \frac{v_x^2}{r} \] Substituting, we find: \[ a_c = \frac{(12.99)^2}{333} \approx 0.506 \text{ m/s}^2 \]

Step 6: Conclusion

The minimum centripetal acceleration required for the SUV to successfully clear the gully is approximately \(0.506 \text{ m/s}^2 \).

Key Concepts

Projectile Motion

Projectile motion often begins with an object in motion having only an initial velocity and then being subjected to gravitational forces alone. In this problem, the SUV must perform a projectile motion to cross the gully. It starts on a circular path, takes off, and is acted upon only by gravity until it lands.

Key principles involved include:

• Horizontal and vertical components of motion are treated separately.
• The horizontal motion is at constant velocity, and vertical motion is subject to acceleration due to gravity.
• The time of flight depends solely on the vertical drop.

For the calculation, the drop height of the gully determines the time in the air using the formula:
\[ h = \frac{1}{2} g t^2 \] This physics of projectile motion helps determine how fast and at what angle the SUV needs to leave the circular path to land safely.

Centripetal Force

Centripetal force is the force that keeps an object moving along a circular path. It is directed toward the center of the circle. For this SUV, while on the curve, the centripetal force is what maintains its circular path until it starts jumping the gully.

In equations, centripetal force is defined as:
\[ F_c = \, \frac{m v^2}{r} \] where:

• \( F_c \) is the centripetal force,
• \( m \) is the mass of the object,
• \( v \) is the tangential velocity, and
• \( r \) is the radius of the circular path.

For the SUV to have the needed speed to jump, it must have enough centripetal force. This is crucial in physics problem-solving as miscalculations can lead to failure of the intended motion.

Circular Motion

When a vehicle moves in a circular path, its velocity is constantly changing direction, which implies it is accelerating, even if it maintains a constant speed. This type of acceleration is called centripetal acceleration. It is directed towards the center of the circle which keeps the vehicle on track.

For calculating centripetal acceleration, the formula is:
\[ a_c = \, \frac{v^2}{r} \] where:

• \( a_c \) is the centripetal acceleration,
• \( v \) is the tangential velocity, and
• \( r \) is the radius of the curvature.

Understanding circular motion is key to determining the minimum centripetal acceleration needed for the SUV. By relating velocity to radius, we can calculate the required speed necessary to 'launch' the vehicle into a successful jump.

Physics Problem Solving

Physics problem-solving is a systematic approach to apply physics concepts to find solutions. In solving the SUV problem, a detailed step-by-step approach was crucial.

Steps to solving such problems include:

• Understanding the problem: Identify known and unknown variables.
• Convert units when necessary to ensure compatibility.
• Use equations of motion to find missing quantities.
• Relate them to other equations that describe real-world constraints (like centripetal acceleration).

In this case, calculating the time to fall using vertical motion equations provided a pathway to determine horizontal velocity. Problem-solving techniques in physics often require breaking down problems into manageable parts, performing calculations, and integrating results to find a solution.