The Problem
A motorist travelling at u ms^-1 joins a straight motorway. On the motorway she travels with a constant acceleration of 0.07 m s2 until her speed has increased by 2.8 m s1. (a) Calculate the time taken for this increase in speed. (b) Given that the distance travelled while this increase takes place is 1050 m, find u. (OCR)
Douglas Quadling Mechanics1 Miscellaneous Exercise1 Q2
Problem Statement
A motorist traveling at u m/s joins a straight motorway. On the motorway, she travels with a constant acceleration of 0.07 m/s² until her speed has increased by 2.8 m/s.
(a) Calculate the time taken for this increase in speed.
(b) Given that the distance traveled while this increase takes place is 1050 m, find u.
Solution
Part (a): Calculate the time taken for the increase in speed
We are given:
- Initial speed, u (unknown)
- Acceleration, a = 0.07 m/s²
- Increase in speed, Δv = 2.8 m/s
The formula for acceleration is:
a = Δv / t
Rearranging for time t:
t = Δv / a
Substitute the given values:
t = 2.8 / 0.07 = 40 seconds
Answer: The time taken is 40 seconds.
Part (b): Find the initial speed u
We are given:
- Distance traveled, s = 1050 m
- Acceleration, a = 0.07 m/s²
- Time, t = 40 seconds (from part (a))
The formula for distance traveled under constant acceleration is:
s = ut + 0.5 * a * t²
Substitute the known values:
1050 = u * 40 + 0.5 * 0.07 * (40)²
Simplify the equation:
1050 = 40u + 56
Solve for u:
40u = 1050 – 56
40u = 994
u = 994 / 40 = 24.85 m/s
Answer: The initial speed u is 24.85 m/s.
