Formula sheet: Motion in a Straight Line

Motion in a Straight Line14 min readLesson 1 of 6

1.1 Definitions that the whole chapter uses Position fixes a point on a chosen axis with an origin and a positive direction. Displacement is the change in position, and path length is the distance actually covered along the route. The two are equal in size only when the motion never reverses. Average velocity divides displacement by the time interval; average speed divides path length by the time interval, so average speed is never smaller than the magnitude of average velocity. Instantaneous velocity is what the average velocity approaches as the time interval shrinks to zero, that is, the derivative of position with respect to time. The magnitude of instantaneous velocity is the instantaneous speed. Acceleration is the rate of change of velocity, so it is the derivative of velocity and the second derivative of position.

v ˉ = \frac{x _{2} - x _{1}}{t _{2} - t _{1}}, v = \frac{d x}{d t}, a = \frac{d v}{d t} = \frac{d ^{2} x}{d t ^{2}}

Average velocity, instantaneous velocity and acceleration

1.2 The kinematic equations The next three equations apply only to motion with constant acceleration. Each one leaves out a different quantity, and choosing the right one is the main skill in the numericals. Always write down the knowns with their signs first. For the distance covered in the nth second, subtract the displacement up to n seconds from the displacement up to n minus one second. The result depends only on the initial velocity and the acceleration.

v = u + a t, x = u t + \frac{1}{2} a t^{2}, v^{2} = u^{2} + 2 a x

Kinematic equations for uniform acceleration, with x measured from the starting point

x_{n} = u + \frac{a}{2} (2 n - 1)

Distance covered in the nth second for constant acceleration

1.3 Reading the graphs On a position-time graph the slope at a point gives the velocity there, and the sign of the slope gives its direction. A straight line means uniform velocity, and a curve with increasing slope means increasing speed. Two position-time curves that cross mean the two objects occupy the same position at that time. On a velocity-time graph the slope gives the acceleration and the area between the curve and the time axis gives the displacement. Areas below the axis count as negative displacement but as positive path length. A velocity that changes sign means the body has reversed direction, and at that instant the velocity is zero while the acceleration is usually not. No object can have two positions or two velocities at the same instant, so a position-time or velocity-time graph can never double back on itself.

x = \int v d t, Δ x = signed area under the v - t graph

Displacement from the area under a velocity-time graph

1.4 Free fall and relative velocity A body released near the Earth falls with a constant acceleration g directed downward when air resistance is neglected. Taking upward as positive, a = -g, and every kinematic equation applies with that single sign change. At the highest point of a vertical throw the velocity is zero and the acceleration is still g downward. Relative velocity compares two motions measured from the same ground frame. The velocity of A relative to B is the ground velocity of A minus that of B. For one dimensional motion the subtraction is enough; for motion along a straight road with both bodies in the same direction the relative speed is the difference, and for opposite directions it is the sum.

v_{A B} = v_{A} - v_{B}, h = \frac{1}{2} g t^{2}, v^{2} = 2 g h

Relative velocity in one dimension and the free fall results from rest

1.5 Units, signs and checklist Displacement is in metre, velocity in metre per second, acceleration in metre per second squared and time in second. Convert km h^-1 to m s^-1 by multiplying by 5/18 before substituting. The sign conventions carry most of the marks: choose an origin, choose a positive direction, and keep every quantity in one frame. Situation Sign of velocity Sign of acceleration Speeding up along positive x positive positive Slowing down along positive x positive negative Moving back toward the origin negative whichever turns it Body at the top of a vertical throw zero negative, equal to -g

Situation	Sign of velocity	Sign of acceleration
Speeding up along positive x	positive	positive
Slowing down along positive x	positive	negative
Moving back toward the origin	negative	whichever turns it
Body at the top of a vertical throw	zero	negative, equal to -g

Exam

The two commonest losses in this chapter are using average speed where average velocity is asked, and forgetting that displacement can be zero while the distance travelled is large. Write the initial and final positions separately before dividing by time.

in hinglish

Is chapter ka base simple hai: position, displacement, velocity aur acceleration . Displacement sirf initial aur final position par depend karta hai, jabki path length poora rasta hai, isliye average speed average velocity ke magnitude se chhoti nahi ho sakti. Instantaneous velocity dx/dt hoti hai aur graph par slope se milti hai. Kinematic equations sirf constant acceleration ke liye hain: v = u + at, x = ut + 1/2 a t^2, v^2 = u^2 + 2ax, aur nth second ke liye u + a(2n-1)/2 . Graph padhna zaroori hai: x-t graph ka slope velocity deta hai, v-t graph ka slope acceleration aur area displacement. Free fall mein a = -g lete hain, aur highest point par velocity zero hoti hai par acceleration g hi rehta hai. Relative velocity v_AB = v_A - v_B hai. Numerical mein pehle signs fix karo, phir equation chuno.