Maths › Mechanics › Kinematics with constant acceleration
Kinematics with constant acceleration
When acceleration holds steady, five quantities lock together and any three determine the other two. The suvat equations are the bookkeeping, the velocity-time graph is the reason they work, and gravity is the standard example.
Builds on Modelling, quantities and units.
Where it earns its keep: Motion graphs and the SUVAT equations on InkPhysics.
IN THIS TOPIC
- Read velocity from a displacement-time graph, and acceleration and displacement from a velocity-time graph.
- Select and use the five suvat equations for constant acceleration.
- Split a multi-stage motion at the moments the acceleration changes.
- Model vertical motion under gravity, using the symmetry of up-and-down flight.
COMMON MISCONCEPTION
At the top of its flight a thrown ball has zero acceleration.
What the graphs are telling you
On a displacement-time graph the gradient is the velocity, so a horizontal stretch means the object has stopped and a straight sloping line means steady speed. On a velocity-time graph the gradient is the acceleration and the area underneath is the displacement. Constant acceleration draws a straight line, and every suvat equation is that trapezium written in symbols.
Multi-stage motion is an area problem and nothing more. A car accelerating from rest to 20 m s⁻¹ in 8 s, holding that for 30 s, then stopping in a further 12 s draws a trapezium with parallel sides 50 and 30 and height 20. Its area is ½(50 + 30) × 20 = 800 m. No equation was needed.
Take care with area below the axis. On a velocity-time graph that area counts as negative displacement, so a question about total distance wants the areas added as magnitudes, while a question about displacement wants them signed.
The five equations
All five are printed in the booklet under Kinematics, so copy them off the page rather than trusting your recall of a sign. Each equation leaves out exactly one of the five letters, and the last one, which drops u, is the one candidates forget they own. The working method is clerical and completely reliable. List s, u, v, a, t down the margin, fill in the three you know, then pick the equation missing the letter you do not care about.
WORKED EXAMPLE
A braking train
A train slows from 24 m s⁻¹ to 12 m s⁻¹ over 300 m. Find the deceleration.
Known: u = 24, v = 12, s = 300. Wanted: a. Time is not involved, so use v² = u² + 2as.
144 = 576 + 600a, so a = −432/600 = −0.72 m s⁻², a deceleration of 0.72.
The sign is negative because the train slows while travelling in the positive direction.
These equations hold while the acceleration is constant and at no other time. If a question changes the forces partway through, stop, take the end of stage one as the start of stage two, and start a fresh suvat list.
Gravity: the standard constant
Free flight near the ground has constant acceleration g = 9.8 m s⁻² downwards, with air resistance modelled away. Take up as positive and a = −9.8 for the whole flight, going up, at the top, and coming down. At the peak the velocity is zero. The acceleration never is.
A ball thrown up at 14.7 m s⁻¹ reaches v = 0 when t = 14.7/9.8 = 1.5 s, at height 14.7 × 1.5 − 4.9 × 1.5² = 11.0 m, and returns to the hand with speed 14.7 again. That symmetry is a free answer-checker, so use it.
ASSESSMENT FOCUS
- Write the suvat list and fill it in before choosing an equation. The choice then makes itself.
- suvat applies only while the acceleration is constant. Split the motion at the moment it changes and never run one equation across the join.
- In vertical motion, declare the positive direction and keep g's sign consistent with it for the entire question.
- Graph questions pay for areas and gradients. Say which of the two you are using.
CHECK YOURSELF
A stone is dropped from rest down a well and hits the water after 2 s. Taking g = 9.8 m s⁻², find the depth and the speed at impact.
Show a hint
u = 0 makes two suvat equations very short.
Show the answer
Depth: s = ½ × 9.8 × 2² = 19.6 m.
Impact speed: v = 9.8 × 2 = 19.6 m s⁻¹.
List s, u, v, a, t, then pick the equation missing the letter you do not need.
Gravity is a constant −9.8 with up positive, all flight long, peak included.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the kinematics with constant acceleration questions page.
CHECK YOUR PROGRESS
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- Read velocity from a displacement-time graph, and acceleration and displacement from a velocity-time graph.
- Select and use the five suvat equations for constant acceleration.
- Split a multi-stage motion at the moments the acceleration changes.
- Model vertical motion under gravity, using the symmetry of up-and-down flight.
Open the full revision checklist to see every objective in the course in one place.