Motion in a Straight Line: common doubts, answered
The questions students ask most often about Motion in a Straight Line, each with a short answer. For the full chapter, read the Motion in a Straight Line notes.
About the chapter
Can you average the initial and final speeds to get average speed?
Only when the acceleration is constant and the body does not turn back during the interval; then the average velocity is (v₀ + v)/2 and its size equals the average speed. If the acceleration changes, if the body reverses, or if a journey has two phases of unequal duration, the shortcut fails. In general, divide the total distance by the total time for average speed, and the total displacement by the total time for average velocity.
Why must all quantities be in the same units in kinematics problems?
Because the equations combine speeds, distances, times and accelerations, and mixing units gives nonsense. A speed in km/h used with g in m s⁻² produces a wrong answer. Convert km/h to m/s by multiplying by 5/18 before substituting, and keep every quantity in SI units throughout the calculation.
Background: position, displacement and distance (prerequisite recap)
Read this section in the notes →What is the difference between distance and displacement?
Distance is the total length of the path travelled, while displacement is the change in position from start to finish, with a direction. Distance is a scalar and can never decrease; displacement is a vector and can be zero even after a long journey. A runner completing one lap of a track covers 400 m of distance but has zero displacement.
Can displacement be greater than distance?
No, the magnitude of displacement can never exceed the distance travelled. The straight line between two points is the shortest possible path, so any real path is at least that long. The two are equal only when the body moves along a straight line without turning back; otherwise the distance is larger.
Instantaneous velocity and speed
Read this section in the notes →Why is average speed not equal to the magnitude of average velocity?
Because average speed divides the total path length by time, while average velocity divides the net displacement by time. If the body reverses direction, the path is longer than the displacement, so average speed comes out larger. They are equal only for motion in one direction along a straight line. For a full round trip, average velocity is zero but average speed is not.
What is the difference between instantaneous speed and instantaneous velocity?
Instantaneous speed is simply the magnitude of instantaneous velocity, v = dx/dt, at a given moment. Over a vanishingly small interval the path and the displacement become the same, so the two always agree in size. Velocity also carries a sign or direction in straight-line motion, while speed is never negative.
Acceleration
Read this section in the notes →Does negative acceleration always mean the body is slowing down?
No. A body slows down only when its acceleration and velocity point in opposite directions. The sign of acceleration depends on which direction you chose as positive. A body moving in the negative direction with negative acceleration is actually speeding up. Always compare the signs of v and a rather than reading the sign of a on its own.
Can a body have zero velocity but non-zero acceleration?
Yes. A ball thrown straight up has zero velocity at its highest point, yet its acceleration is still g downward. Acceleration measures how fast velocity is changing, not the velocity itself, and at the top the velocity is changing from upward to downward. If the acceleration were zero there, the ball would simply hang in the air.
Reading motion graphs
Read this section in the notes →What does the slope of a position-time graph tell you?
The slope of an x-t graph at any point gives the instantaneous velocity at that moment. A straight line means constant velocity, a steeper line means faster motion, and a downward slope means motion in the negative direction. A curved graph means the velocity is changing, so the body is accelerating.
What does the area under a velocity-time graph give?
The area between a v-t graph and the time axis gives the displacement over that interval. Area below the axis counts as negative because the body is moving in the negative direction then. To get distance instead, add all the areas as positive values. The slope of the same graph gives the acceleration.
Why can't a position-time graph have two values at the same time?
Because a body cannot be in two places at one instant, so a vertical line through any time must cross a real x-t graph only once. For the same reason a speed-time graph cannot dip below zero, and a real velocity cannot jump instantly from one value to another, since that would need infinite acceleration.
Kinematic equations for uniformly accelerated motion
Read this section in the notes →When can you use the equations v = v₀ + at and v² = v₀² + 2ax?
Only when the acceleration is constant in both size and direction throughout the interval. If acceleration changes with time, as in x = 6t² − t³, these equations give wrong answers and you must use calculus: v = dx/dt and a = dv/dt. Also make sure the motion is along a single straight line before applying them.
Why does x = v₀t + ½at² not give the distance when the body turns back?
Because that equation gives displacement, the net change in position, not the length of the path. If the velocity reverses during the interval, part of the motion cancels the rest. To get the distance, find the time when v becomes zero, compute the displacement before and after that instant separately, and add their magnitudes.
How do you find the distance covered in the nth second?
Use sₙ = v₀ + a(2n − 1)/2, which is the displacement during the interval from t = n − 1 to t = n seconds. It comes from subtracting the displacement in n − 1 seconds from that in n seconds. Note that the result is in metres even though it looks like a velocity, because it is multiplied by an implied one second.
Free fall and vertical motion under gravity
Read this section in the notes →Do heavier objects fall faster than lighter ones?
No. Without air resistance every object near the Earth's surface falls with the same acceleration g, about 9.8 m s⁻², whatever its mass. A feather falls slowly in air only because drag is large compared with its tiny weight. In a vacuum a feather and a coin dropped together reach the ground at the same time.
What sign should g have in vertical motion problems?
Pick one direction as positive and keep it for the whole motion. If upward is positive, then g is −9.8 m s⁻² both while the ball rises and while it falls, because gravity always pulls downward. Switching the sign of g midway is a common error; with one fixed sign convention a single equation handles the whole flight.
Why is the time to go up equal to the time to come down?
Without air resistance the motion is symmetric: the ball loses speed at rate g going up and gains it at the same rate coming down. So it takes u/g to reach the top and u/g to fall back, and it returns to the launch point with the same speed u. Air drag breaks this symmetry, making the descent take longer.
Stopping distance and reaction time
Read this section in the notes →Why does doubling the speed make the stopping distance four times larger?
Because the braking distance is d = v₀²/2a, which depends on the square of the initial speed. For the same braking deceleration, a car at twice the speed needs four times the distance to stop. This is why speed limits matter so much on roads: a modest increase in speed greatly lengthens the distance needed to stop.
How is reaction time measured with a falling ruler?
A friend drops a ruler between your fingers and you catch it as soon as you see it move. The distance d it falls before you grab it gives your reaction time through t = √(2d/g), because the ruler starts from rest in free fall. A typical value is a few tenths of a second.
Using calculus in kinematics
Read this section in the notes →When should you use calculus instead of kinematic equations?
Use calculus whenever the acceleration is not constant, for example when position or velocity is given as a function of time. Differentiate x(t) to get v = dx/dt, and differentiate again for a = dv/dt. Going the other way, integrate the acceleration to get velocity and integrate velocity to get displacement, using the initial values to fix the constants.
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