JEE Physics Vectors and Calculus Notes Set 01

Scalars and Vectors

A scalar has magnitude only (mass, distance, speed, time, energy). A vector has magnitude and direction and adds by the triangle or parallelogram law (displacement, velocity, force). Some quantities have both a size and a direction but are not vectors, for example current, because they do not obey vector addition.
1. Equal vectors have the same magnitude and direction. The negative of a vector has the same magnitude and opposite direction.
2. A unit vector is \( \hat A = \frac{\vec A}{|\vec A|} \). A null vector has zero magnitude and no definite direction.
3. Multiplying by a scalar \( \lambda \) gives a vector of magnitude \( |\lambda||\vec A| \), along \( \vec A \) if \( \lambda > 0 \) and opposite if \( \lambda < 0 \).
4. Position vector of a point (x, y, z) is \( x\hat i + y\hat j + z\hat k \). The vector from point P to point Q is the position vector of Q minus that of P.

Addition and Subtraction

If two vectors of magnitude A and B make an angle \( \theta \), their resultant has magnitude \( R = \sqrt{A^2 + B^2 + 2AB\cos\theta} \) and makes an angle \( \alpha \) with \( \vec A \) given by \( \tan\alpha = \frac{B\sin\theta}{A + B\cos\theta} \).
1. Maximum resultant is \( A + B \) at \( \theta = 0^\circ \); minimum is \( |A - B| \) at \( \theta = 180^\circ \). Any value in between is possible, and three vectors can have zero resultant only if the largest does not exceed the sum of the other two.
2. The difference \( \vec A - \vec B \) is \( \vec A + (-\vec B) \), with magnitude \( \sqrt{A^2 + B^2 - 2AB\cos\theta} \).
3. For equal vectors, \( R = 2A\cos\frac\theta2 \), along the bisector.
4. Lami's theorem: three forces in equilibrium satisfy \( \frac{P}{\sin\alpha} = \frac{Q}{\sin\beta} = \frac{R}{\sin\gamma} \), where each angle lies opposite its force, that is between the other two forces.
5. A closed polygon of vectors taken in order has zero resultant.

Components

A vector at angle \( \theta \) with the x-axis has components \( A_x = A\cos\theta \) and \( A_y = A\sin\theta \). Then \( A = \sqrt{A_x^2 + A_y^2} \) and \( \tan\theta = A_y/A_x \). In three dimensions \( \vec A = A_x\hat i + A_y\hat j + A_z\hat k \), and the direction cosines \( l, m, n \) satisfy \( l^2 + m^2 + n^2 = 1 \).
The component of \( \vec A \) along \( \vec B \) is \( \vec A\cdot\hat B = A\cos\theta \). To add vectors, add their components axis by axis. The x-component of a resultant is the sum of the x-components of the parts.

Dot Product and Cross Product

The magnitude of the cross product is the area of the parallelogram formed by the two vectors. Also \( (\vec A\cdot\vec B)^2 + |\vec A\times\vec B|^2 = A^2B^2 \). A right-handed coordinate system obeys \( \hat i\times\hat j = \hat k \).

Functions and Derivatives

If y depends on x through a rule \( y = f(x) \), the derivative is the instantaneous rate of change, \( \frac{dy}{dx} = \lim_{\Delta x\to0}\frac{\Delta y}{\Delta x} \), the slope of the tangent to the graph.

Rules: sum rule \( (u + v)' = u' + v' \); constant multiple \( (cu)' = cu' \); product rule \( (uv)' = u'v + uv' \); quotient rule \( \left(\frac uv\right)' = \frac{u'v - uv'}{v^2} \); chain rule \( \frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx} \), used for functions like \( \sin5x \), \( (2x+1)^5 \) or \( e^{2x} \).

Derivatives in Physics

Velocity is \( v = \frac{dx}{dt} \) and acceleration is \( a = \frac{dv}{dt} = \frac{d^2x}{dt^2} = v\frac{dv}{dx} \). Force is \( F = \frac{dp}{dt} \), current is \( i = \frac{dq}{dt} \), angular velocity is \( \omega = \frac{d\theta}{dt} \). In each case the derivative is taken first and the particular time is substituted afterwards.

Maxima and Minima

At a maximum or a minimum of \( y = f(x) \), the slope is zero, \( \frac{dy}{dx} = 0 \).
1. If \( \frac{d^2y}{dx^2} < 0 \) at that point, it is a maximum (the slope changes from positive to negative).
2. If \( \frac{d^2y}{dx^2} > 0 \), it is a minimum.
3. For a problem with a constraint (such as a fixed area of sheet), first use the constraint to write the quantity as a function of one variable, then differentiate.
For a particle moving along a line, velocity is zero at the extreme positions, and speed is extreme where acceleration is zero.

Integration

Integration reverses differentiation. The indefinite integral \( \int f(x)dx = F(x) + C \) where \( F' = f \).

A definite integral is \( \int_a^b f(x)dx = F(b) - F(a) \). It equals the area under the curve between x = a and x = b when the curve lies above the axis; area below the axis counts as negative. Practical uses: displacement \( = \int v\,dt \), change in velocity \( = \int a\,dt \), work \( = \int F\,dx \), charge \( = \int i\,dt \).

Worked Example 1

A particle moves with \( x = t^3 - 6t^2 + 3t + 7 \). Find its velocity when its acceleration is zero.
\( v = 3t^2 - 12t + 3 \) and \( a = 6t - 12 = 0 \), so \( t = 2 \) s. Then \( v = 12 - 24 + 3 = -9 \) m/s.

Worked Example 2

Forces \( \vec F_1 = 2\hat i + 3\hat j \) and \( \vec F_2 = \hat i - \hat j + 2\hat k \) act on a body that moves through \( \vec d = 3\hat i + \hat k \). Find the total work.
Resultant \( \vec F = 3\hat i + 2\hat j + 2\hat k \). Work \( = \vec F\cdot\vec d = 3(3) + 0 + 2(1) = 11 \) units.