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Consider the angle of inclination to be a and the angle of projection to be b with respect to the ground. Also let the velocity of projection be V. Consider X axis along the incline and Y axis perpendicular to the incline plane.

Therefore V_x = V cos (b - a) and V_y = V Sin (b - a)

g_x = g Sin (a) and g_y = g cos (a)

Y = V_y t - 0.5 g_y t ²

X = V_x t - 0.5 g_x t ²

Solving the equations we get the required results.

Time of flight can be obtained by putting Y = 0;

After obtaining T the time of flight, substitute T in X and obtain the range. Check for its maximum value.

A projectile projected at an angle q with the horizontal and velocity u has a horizontal velocity u cos q and vertical velocity u sin q. During collision with the ground only the vertical component undergoes a change because, it is perpendicular to the surface at which it is colliding. The horizontal component does not undergo any change because it is parallel to the surface.

The vertical component becomes e usin q after the first collision and e² usin q after the second collision and so on. Therefore after n^th collision the vertical component becomes

V₁ = eⁿ u sin q -------------(1)

And the horizontal component remains

V₂ = u cos q --------------(2)

Therefore the magnitude of the velocity after n^th collision will be

V_n = Ö V₁² + V₂² ---------------- (3)

And its direction of motion with the horizontal will be

a = Tan -¹ ( V₁ / V₂) ------------------ (4)

Intuitive Definition. Let y = f(x) be a function. Suppose that a and L are numbers such that

o whenever x is close to a but not equal to a, f(x) is close to L;

o as x gets closer and closer to a but not equal to a, f(x) gets closer and closer to L; and

o suppose that f(x) can be made as close as we want to L by making x close to a but not equal to a.

Then we say that the limit of f(x) as x approaches a is L and we write

limit x --> a f (x) = L

Q. a rod of mass m,length lis allowed to stand on a smooth surface.it is given a horizontal force at the top most point .what would be the angular velocity

when the rod is about to touch the ground

The question is not clear whether the rod is applied by the force continuously or momenterily.

Since the torque about the vertical axis is zero, we can conserve angular momentum about the vertical axis.

m v d = I w + m d² w -------------- (1)

Where 'm' is the mass of the bullet, 'd' is the distance of the center of the door from the axis, 'w' is the angular speed after impact, 'v' is the velocity of the bullet before impact, 'I' is the moment of inertia of the door about the axis of rotation.

The equation for the equilibrium of the metal piece is

N + F = m g ------------------ (1)

Where F is the buoyant force and N the normal reaction.

F = mg x density of the liquid / density of the solid = mg / 8

Therefore

N = mg - F = mg - mg / 8 = 7 mg / 8.

Identify the roots of a quadratic equation from the graph and the factored equation.
Understand connections between coefficients of the second and third terms of a quadratic equation and the roots of the equation.

Students learn to factor equations but often they don't have the conceptual understanding to accompany what they do mechanically with the numbers.

Given the equation of a parabola: y = x² - bx + c we start with an example with roots 2 and -3. Our equation is y = x² + x - 6.

Concepts to be emphasized during the activity include:

· 2 + (-3) = -1 or the coefficient of the second term, -b

· (2)(-3) = -6 or the coefficient of the third term, c

· The sum of the two numbers is -1 and the product of the two numbers is -6.

· When a quadratic equation is graphed, it is a parabola

· The roots satisfy the equation so that y equals zero and therefore, most importantly, the roots of the equation can be read from the graph where the lines of the parabola cross the x axis.

Each stall can be loaded with either a cow or a calf or a horse. Therefore each stall can be loaded in three ways. Therefore the total no. of ways is

3¹²

You have given the answer but neither in the question nor in the answer you have mentioned what is K? If K is the mass per unit length then only the answer given by you will match with velocity units. If you say that it is so then your answer is lacking the relation between velocity and hanging length ( x) of the chain.

And if you say that k is length of the hanging portion of the chain then the units will not match with that of the velocity!

Now it is your turn to verify what is correct!

A heap of chain is lying on a horizontal table.A small section of the chain is released through a hole in the table. calculate the velocity as a function of the length of the chain hanging vertically.

sorry!!this is not the given answer.The correct answer is (2mg/k)^1/2

Question is either incomplete or not understandable.

If | sin x | = | x - k| , then as 0 £ | sin x | £ 1, we have

0 £ | x - k| £ 1.

To have no real solution

If k £ x

x - k ³ 1, then K £ x - 1;

If k ³ x

K - x ³ 1, then k ³ x + 1;

Since each term is a perfect square, each term should be individually zero.

If a₁ / a₂ ¹ b₁ / b₂ then there are 'n' solutions.

X₁ = - b₁ / a₁ ; X₂ = - b₂ / a₂ ; ......... ; X_n = - b_n / a_n ;

If a₁ / a₂ = b₁ / b₂ , then there is only one solution

X = - b₁ / a₁ = - b₂ / a₂ = ............... = - b_n / a_n

Two Cases arise:

Case 1: When the conductor is not immersed in an External Electric Field and there is no current flow or rather under electrostatic conditions.

When there is no external electric field the total charge inside the conductor is zero. Hence there is no electric field inside the conductor. If we argue that there are electrons and protons, then we say that the lines of force start at positive charge and end at an equal amount of negative charge. Therefore the electric field exists inside an atom but not outside it. Since the positive and negative charge inside a conductor is equal, we conclude that the field inside a conductor is zero.

Case 2. When the conductor is immersed in an External Electric Field and there is no current flow or rather under electrostatic conditions.

When external electric lines fall on a conductor negative charges flow opposite to the lines of force and accumulate on one side of the conductor and positive charge accumulates on the opposite side of negative charges. These are called induced charges. These induced charges produce an electric field inside the conductor and opposite to the external electric field. The two fields being equal in magnitude and opposite cancel. Therefore the electric field inside a conductor is zero.

When a bob in a simple pendulum is under the action of another force like that of buoyant force or electrostatic force, we use the following formula.

T = 2p Ö l / (g + a) , Where g and a are vector quantities and further a magnitude is given by F / m , where F is the force acting on the bob other than due to gravity. In this problem we have

F = m r g / s => F / m = r g / s

Since the direction of this force is opposite to that of g we get the time period of the pendulum as,

T = 2p Ö l / (g - (r g / s))