🔑:To analyze the motion of the snow from the moment it leaves the shovel to when it lands on the snowbank, we'll apply Newton's three laws of motion, considering all forces acting on the snow, including gravity and air resistance.## Step 1: Identify the Forces Acting on the SnowWhen the snow leaves the shovel, it is subject to two primary forces: gravity (acting downward) and air resistance (acting opposite to the direction of motion). Initially, when thrown, the snow also has an upward and possibly forward component of velocity imparted by the person throwing it.## Step 2: Apply Newton's First Law of MotionNewton's first law states that an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force. As soon as the snow leaves the shovel, it is no longer in uniform motion due to the external forces acting upon it, primarily gravity and air resistance.## Step 3: Analyze the Effect of GravityGravity acts downward on the snow with a force equal to the mass of the snow multiplied by the acceleration due to gravity (approximately 9.81 m/s^2 on Earth). This force causes the snow to accelerate downward.## Step 4: Consider the Impact of Air ResistanceAir resistance acts in the opposite direction to the motion of the snow. Its effect depends on the velocity of the snow and the density of the air. For a piece of snow, air resistance can significantly affect its motion, especially if it is thrown with a low initial velocity or if the air is dense.## Step 5: Apply Newton's Second Law of MotionNewton's second law states that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to its mass. The equation is F = ma, where F is the net force, m is the mass of the object, and a is the acceleration. In this scenario, the net force acting on the snow is the vector sum of gravity and air resistance. The acceleration of the snow is thus determined by the balance between these forces and its mass.## Step 6: Describe the Acceleration of the SnowInitially, the snow may have an upward and forward acceleration due to the force imparted by the shovel. However, as it rises, gravity begins to dominate, causing the snow to decelerate in the upward direction and eventually accelerate downward. Air resistance affects both the upward and downward motion, slowing the snow down. The forward motion, if any, is also affected by air resistance, causing the snow to lose speed over time.## Step 7: Consider Newton's Third Law of MotionNewton's third law states that for every action, there is an equal and opposite reaction. When the person throws the snow, the force exerted on the snow is matched by an equal and opposite force exerted on the person (though the effect on the person is usually negligible due to their larger mass compared to the snow).The final answer is: boxed{F = ma}

🔑:## Step 1: Understanding Newton's 2nd LawNewton's 2nd Law states that the acceleration of an object is directly proportional to the net force acting upon the object and inversely proportional to its mass. The formula is F = ma, where F is the net force applied to the object, m is the mass of the object, and a is the acceleration.## Step 2: Applying Newton's 2nd Law to the Car ScenarioIf the net force on the car is zero, according to Newton's 2nd Law (F = ma), the acceleration (a) must also be zero because F = 0. When acceleration is zero, the velocity of the object remains constant. Therefore, the car would maintain its velocity and not slow down.## Step 3: Analyzing the Acceleration and Force RelationshipFor the second part of the question, if two objects have the same acceleration, it means they are changing their velocities at the same rate. However, according to Newton's 2nd Law, the force required to achieve a certain acceleration depends on the mass of the object (F = ma). Therefore, if two objects have different masses but the same acceleration, they must be under the influence of different forces, as the force (F) would need to be adjusted to compensate for the difference in mass (m) to achieve the same acceleration (a).## Step 4: Providing Examples- Example for the car: A car cruising at 60 km/h on a straight, flat road with no net force acting on it (no acceleration) will maintain its velocity of 60 km/h.- Example for objects with the same acceleration: Consider two objects, one with a mass of 2 kg and the other with a mass of 4 kg. If both are to achieve an acceleration of 2 m/s^2, the forces applied must be different. For the 2 kg object, F = ma = 2 kg * 2 m/s^2 = 4 N. For the 4 kg object, F = ma = 4 kg * 2 m/s^2 = 8 N. Thus, the 4 kg object requires twice the force to achieve the same acceleration as the 2 kg object.The final answer is: boxed{0}

🔑:_Solution__Step 1. Plotting the data._To plot the data, we need to decide which variable to use as the independent variable and which variable to use as the dependent variable. In this example, the dependent variable is the monthly consumption and the independent variable is the monthly income. Using the given data, we can plot the data as follows:_Step 2. Estimating the regression model._To estimate the regression model, we need to calculate the means of (X) and (Y), the deviations from the mean, the product of the deviations, and the squares of the deviations.begin{tabular}{c c c c c c} hline hline (X) & (Y) & (X-overline{X}) & (Y-overline{Y}) & ((X-overline{X})(Y-overline{Y})) & ((X-overline{X})^{2}) hline400 & 250 & (-)50 & (-)28.6 & 1428.57 & 2500 420 & 270 & (-)30 & (-)8.6 & 258 & 900 450 & 300 & (-)10 & 21.4 & (-)214 & 100 500 & 350 & 40 & 71.4 & 2856 & 1600 550 & 380 & 90 & 101.4 & 9130.6 & 8100 hline (Sigma X=2320) & (Sigma Y=1550) & & & (Sigma(X-overline{X})(Y-overline{Y})=13410.23) & (Sigma(X-overline{X})^{2}=13100) (overline{X}=2320/5=464) & (overline{Y}=1550/5=310) & & & & hline hline end{tabular}Using the formula for the slope of the regression line,[b_{1}=frac{Sigma(X-overline{X})(Y-overline{Y})}{Sigma(X-overline{X})^{ 2}}=frac{13410.23}{13100}=1.0237]Using the formula for the intercept of the regression line,[b_{0}=overline{Y}-b_{1}overline{X}=310-1.0237(464)=88.55]Hence, the estimated regression model is[Y=88.55+1.0237X]_Step 3. Using the model to predict consumption._To predict the consumption when the income is 500, we use the estimated regression model as follows:[Y=88.55+1.0237(500)=88.55+511.85=600.4]

🔑:## Step 1: Understand the ProblemThe problem involves a sphere with a non-uniform diameter subjected to uniform pressure. We need to derive a formula for the required strength of the sphere, considering the variance in radius and thickness.## Step 2: Identify Key FactorsThe key factors in this problem are the radius 'r', thickness 'T', water pressure 'Pw', and internal pressure 'Pi'. The sphere's strength will depend on its ability to withstand the pressure difference between the inside and outside.## Step 3: Calculate Pressure DifferenceThe pressure difference (ΔP) between the inside and outside of the sphere is given by ΔP = Pw - Pi.## Step 4: Consider BuoyancySince the sphere is immersed in a fluid and has mass pumped out to create buoyancy, we need to consider the effect of buoyancy on the sphere's strength. However, buoyancy affects the net force on the object rather than its material strength.## Step 5: Derive Formula for Required StrengthThe required strength of the sphere can be related to the pressure difference and its geometry. For a sphere, the stress (σ) due to internal pressure is given by σ = ΔP * r / (2 * T), assuming a thin-walled sphere. This formula provides a basis for calculating the required strength.## Step 6: Account for Non-Uniform DiameterGiven the non-uniform diameter, we must consider how variations in radius and thickness affect the sphere's strength. However, without specific details on the variance, we can generalize that the formula must accommodate the minimum thickness and maximum radius to ensure the sphere can withstand the pressure difference.## Step 7: Finalize FormulaConsidering the above steps, the formula for the required strength (σ) of the sphere, taking into account the variance in radius and thickness, can be generalized as σ = ΔP * r_max / (2 * T_min), where r_max is the maximum radius and T_min is the minimum thickness.The final answer is: boxed{frac{Delta P r}{2T}}