For a scene in a movie, a stunt driver drives a 1.40x10^3 kg pickup truck with a length of 4.53 m around a circular curve with a radius of curvature of 0.333 km. The truck is to curve off the road, jump across a gully 10.0 m wide, and land on the other side 2.96 m below the initial side. What is the minimum centripetal acceleration the truck must have in going around the circular curve to clear the gully and land on the other side?
To find the minimum centripetal acceleration the truck must have to clear the gully and land on the other side, we can start by breaking down the problem into smaller steps:
Step 1: Calculate the speed required to jump the gully.
- Since we know the width of the gully (10.0 m) and the height difference (2.96 m), we can use the equations of projectile motion to find the minimum speed required to clear the gully.
- Using the formula for the horizontal distance (range) of a projectile: R = (v^2 * sin(2θ)) / g, where R is the range, v is the velocity, θ is the launch angle, and g is the acceleration due to gravity.
- Since the truck is launching horizontally, the angle θ is 0 degrees, and sin(0) = 0. Therefore, the range equation simplifies to R = (v^2 * sin(0)) / g = (v^2 * 0) / g = 0.
- To clear the gully, the minimum speed required should ensure that the horizontal distance traveled (range) is at least equal to the width of the gully. Therefore, v^2 = (g * R) / sin(2θ), where v^2 is the minimum required velocity^2, g is the acceleration due to gravity (9.8 m/s^2), R is the width of the gully (10.0 m), and sin(2θ) = sin(0) = 0.
- Therefore, v^2 = (9.8 m/s^2 * 10.0 m) / 0 = 0. The minimum required velocity is 0 m/s. This means the truck cannot clear the gully and land on the other side based on the given parameters.
Step 2: Determine the minimum centripetal acceleration needed to go around the circular curve.
- In this case, we can see that the truck cannot clear the gully, and therefore, it won't reach the circular curve.
- Therefore, the minimum centripetal acceleration is not applicable since the truck cannot even reach the circular curve.
In conclusion, based on the provided parameters, it is impossible for the truck to clear the gully and land on the other side.