Motion in a Plane: common doubts, answered
The questions students ask most often about Motion in a Plane, each with a short answer. For the full chapter, read the Motion in a Plane notes.
About the chapter
Why do we ignore air resistance in projectile and circular motion problems?
Because it makes the motion simple enough to solve exactly, and for compact, dense objects at moderate speeds its effect is small. With drag, the horizontal velocity would fall, the path would no longer be a parabola, and the range would be shorter. The ideal results are good first estimates rather than exact predictions for real throws.
Scalars and vectors
Read this section in the notes →What is the difference between a scalar and a vector?
A scalar is fully described by a number with a unit, while a vector also needs a direction and must add by the triangle or parallelogram law. Mass, time and energy are scalars; displacement, velocity, force and acceleration are vectors. Having a direction is not enough on its own: a quantity is a vector only if it obeys the vector addition rules.
Is electric current a vector?
No, current is a scalar even though it seems to have a direction along a wire. Currents meeting at a junction add as ordinary numbers, not by the parallelogram law, so current fails the defining test of a vector. This is the standard example showing that direction alone does not make a quantity a vector.
Multiplying a vector by a number
Read this section in the notes →What happens when you multiply a vector by a negative number?
The result is a vector whose size is the magnitude multiplied by the absolute value of the number, but pointing in the opposite direction. Multiplying A by −2 gives a vector twice as long as A, reversed. Multiplying by a dimensional scalar, such as time, can also change what the vector represents, as when velocity times time gives displacement.
Does a unit vector have a unit?
No. A unit vector has magnitude one and no unit or dimension at all; its only job is to point in a direction. It is found by dividing a vector by its own magnitude. So î, ĵ and k̂ carry no units, and the unit of a vector such as 3î m belongs to the number 3, not to î.
Adding and subtracting vectors graphically
Read this section in the notes →Why can't you just add the magnitudes of two vectors?
Because the size of the sum depends on the angle between them. The magnitudes add directly only when the vectors point the same way. For 3 m and 4 m at right angles, the resultant is 5 m, not 7 m. In general R² = A² + B² + 2AB cos θ, so the resultant can be anywhere from A − B to A + B.
What is the difference between the triangle law and the parallelogram law of vector addition?
They give exactly the same resultant; only the drawing differs. In the triangle law you place the tail of the second vector at the head of the first and join the free ends. In the parallelogram law you draw both vectors from one point, complete the parallelogram, and take its diagonal from that point. Choose whichever suits the diagram.
How do you subtract one vector from another?
To find A − B, add A to the negative of B, which is B reversed. So A − B = A + (−B), and the triangle or parallelogram law then applies as usual. Vector subtraction is not commutative: B − A has the same magnitude as A − B but points the opposite way.
Resolving vectors into components
Read this section in the notes →When do you use sin and when cos to resolve a vector?
The component along the axis that makes angle θ with the vector uses cos θ; the component along the other axis uses sin θ. If θ is measured from the x-axis, Ax = A cos θ and Ay = A sin θ. If the angle is given from the y-axis, the roles swap. Always check where the angle is measured from instead of using a fixed habit.
Adding vectors analytically
Read this section in the notes →Why is adding vectors by components easier?
Because components along the same axis are just numbers that add algebraically. Add all the x-components to get Rx and all the y-components to get Ry, then find the magnitude from √(Rx² + Ry²) and the direction from tan θ = Ry/Rx. This avoids messy geometry when three or more vectors are involved.
What is the maximum and minimum resultant of two vectors?
The largest resultant is A + B, when the vectors point the same way, and the smallest is the difference of their magnitudes, when they point opposite ways. Any other angle gives a value between these limits. So two forces of 5 N and 3 N can never combine to give 1 N or 9 N.
Velocity and acceleration in a plane
Read this section in the notes →Why is the velocity always tangent to the path?
Velocity is the limit of displacement divided by time as the interval shrinks to zero. As the two positions get closer, the chord joining them turns into the tangent to the path at that point. So the instantaneous velocity always points along the tangent in the direction of motion, even on a curved path.
Constant acceleration in a plane
Read this section in the notes →Can motion in a plane be treated as two separate straight-line motions?
Yes, when the acceleration is constant. The x and y motions are independent, so you can write v = v₀ + at and r = r₀ + v₀t + ½at² for each component separately. Time is the common link between them. This is exactly the method used for projectile motion, where horizontal and vertical parts are solved apart.
Projectile motion
Read this section in the notes →Is the velocity of a projectile zero at the highest point?
No. Only the vertical component is zero at the top; the horizontal component v₀ cos θ₀ is unchanged throughout the flight because nothing acts horizontally when air resistance is ignored. So the speed at the top equals v₀ cos θ₀. It is zero only for a ball thrown straight up.
Which angle of projection gives the maximum range?
On level ground, 45° gives the maximum range, because R = v₀² sin 2θ₀ / g is greatest when sin 2θ₀ = 1. The maximum range is then v₀²/g. Angles that add up to 90°, such as 30° and 60°, give the same range, though the steeper throw rises higher and stays in the air longer.
Does a ball thrown horizontally take longer to fall than one dropped?
No, both reach the ground at the same time if they leave from the same height. The horizontal velocity does not affect the vertical motion, and both start with zero vertical velocity under the same acceleration g. The thrown ball just lands farther away.
What is the shape of a projectile's path and why?
It is a parabola, y = x tan θ₀ − gx² / (2v₀² cos² θ₀). The horizontal distance grows linearly with time, while the vertical position has a term in t². Eliminating t between the two makes y a quadratic in x, which is a parabola. Air resistance distorts this shape, so the result holds only when drag is ignored.
Uniform circular motion
Read this section in the notes →Is there acceleration in uniform circular motion if the speed is constant?
Yes. Velocity is a vector, and in circular motion its direction keeps changing, so the body accelerates even at constant speed. This centripetal acceleration points towards the centre and has magnitude a_c = v²/R = ω²R. It changes only the direction of motion, never the speed.
Is centripetal acceleration constant in uniform circular motion?
Its magnitude v²/R is constant, but its direction is not, because it always points towards the centre and so keeps turning as the body moves. A vector with changing direction is not a constant vector. That is also why the kinematic equations for constant acceleration cannot be used along a circular path.
What is the relation between linear speed and angular speed?
They are related by v = Rω, where R is the radius and ω is the angular speed in radians per second. For a body going round once in time T, ω = 2π/T = 2πν, where ν is the frequency. Points farther from the centre of a rotating disc have a greater linear speed even though all share the same ω.
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