Question
A car accelerates uniformly from rest to a velocity of 30 m/s in 15 seconds. Find the acceleration and the total distance covered.
Quick Answer
Initial velocity u = 0 m/s
Final velocity v = 30 m/s
Time t = 15 s
Acceleration a = (v – u) / t = (30 – 0) / 15 = 2 m/s²
Distance s = (u + v) / 2 × t = (0 + 30) / 2 × 15 = 225 m
The car’s acceleration is 2 m/s², and the total distance covered in this time is 225 metres.
Understanding the question with detailed explanations
A car accelerates uniformly from rest until it reaches a velocity of 30 metres per second in a time of 15 seconds. We are required to calculate two things: the acceleration of the car and the total distance it travels during this time. Since the car starts from rest, its initial velocity is zero, and the increase in speed is steady, this is a simple case of uniform acceleration where the equations of motion can be applied directly.
This type of problem represents real-life scenarios such as a car pulling away from a stoplight and steadily gaining speed. The calculations help us understand how acceleration affects both velocity and displacement over time, which is one of the foundations of kinematics in physics.
The problem gives us three key pieces of data: initial velocity, final velocity, and time. With these, we can calculate acceleration, which is defined as the rate of change of velocity. Since the motion is uniform, acceleration remains constant throughout the 15 seconds. Once acceleration is determined, we can move on to the distance travelled, which depends on both initial and final velocities.
The logic here is straightforward. The acceleration tells us how quickly the car gains speed, while the distance formula allows us to calculate how far the car has gone while its speed changes steadily. The use of average velocity in the distance formula is particularly important because the speed is not the same at the start and the end of the motion.
A glimpse of the final answer
The calculated acceleration is 2 m/s². This means that every second, the car’s velocity increases by 2 metres per second. Over 15 seconds, this steady increase carries the car from rest to 30 m/s, which is consistent with the data provided in the problem.
The total distance travelled is 225 metres. This makes sense because the car covers less distance in the early seconds while moving slowly, but as it speeds up, it covers more ground each second. Using average velocity to calculate the distance ensures we capture this varying rate correctly.
Data
From the problem:
- Initial velocity, u = 0 m/s
- Final velocity, v = 30 m/s
- Time, t = 15 s
What we need:
- Acceleration, a = ?
- Distance covered, s = ?
All values are already in SI units, so no conversions are required.
Formula
The relevant equations of motion are:
- a = (v – u) / t
- s = (u + v) / 2 × t
The first equation defines acceleration under uniform conditions. The second equation gives displacement by multiplying the average velocity by time. Both are chosen here because they directly use the known values and lead to the required results efficiently.
Since the motion is uniformly accelerated, these formulas apply without modification. If acceleration were non-uniform, we would need calculus, but here the situation is simple and direct.
Solution (solving the problem)
Step 1: Find the acceleration.
a = (v – u) / t
a = (30 – 0) / 15
a = 30 / 15
a = 2 m/s²
Step 2: Find the distance covered.
s = (u + v) / 2 × t
s = (0 + 30) / 2 × 15
s = 30 / 2 × 15
s = 15 × 15
s = 225 m
Final Answer
The car’s acceleration is 2 metres per second squared (2 m/s²).
The total distance covered during the 15 seconds of motion is 225 metres.
Helpful Explanation
This result shows how quickly a car can pick up speed under steady acceleration. A rate of 2 m/s² is reasonable for many vehicles, and reaching 30 m/s (108 km/h) in 15 seconds is consistent with a strong but realistic acceleration. The distance of 225 metres also reflects how much ground is covered while building up speed, which is important in understanding stopping and starting distances in driving.
A common mistake students make is forgetting that the car starts from rest (u = 0), which can lead to errors if they try to use the wrong formula. Another pitfall is confusing average velocity with final velocity when calculating distance. The safe approach is to always write down the known values, select the appropriate equation of motion, and substitute carefully.