Car and bicycle collision problem

Ishfa · Sep 6, 2025

Handwritten derivation integrating a = -c(t - t1) to get dv0 = -c(t - t1) dt, with limits from v0 to v and t1 to t, leading to -v/c + v0/c = t^2/2 - t1 t + t1^2/2 and a final line - v/c = -v0/c + t^2/2 - t1 t + t1^2/2.

Handwritten derivation on lined paper of a piecewise velocity function v_e equal to v_0 for t less than t_1, and v_0 minus c/2 times (t minus t_1) squared for t_1 less than t less than t_2, followed by integration to obtain x_e = v_0 t minus c/6 times (t minus t_1) cubed.

Printed physics problem 'Car and Bicycle Rider': a car at speed v0 slows to a stop, with piecewise acceleration a_c = 0 for 0 < t < t1 and -c(t - t1) for t1 < t < t2, followed by part (a) asking for v_c(t) and x_c(t).

PeroK · Sep 6, 2025

Ishfa said:

Homework Statement: Is the answer, xc = v0t -c( t - t1 )^3 /6 or xc(t)=v0t1+v0(t−t1)−c(t−t1)^3 /6 ? Which one is correct and why?

Those two expressions are equal!

kuruman · Sep 6, 2025

The expressions may be equal, but neither of them can be considered the answer. Variable "t" in these expressions is "any time t". How do these answers reflect the fact that the car stops at t = t₂?
Hint: x_c(t) is a piecewise continuous function.

PeroK · Sep 6, 2025

kuruman said:

Hint: x_c(t) is a piecewise continuous function.

It's actually a continuous function.

PeroK · Sep 6, 2025

PS it's actually a Spline function - continuous and piecewise polynomial in ##t##.

kuruman · Sep 6, 2025

PeroK said:

PS it's actually a Spline function - continuous and piecewise polynomial in ##t##.

I defer the appellation details to you. I just wanted to convey the "piecewise" and "continuous" ideas to the OP.

nasu · Sep 6, 2025

What's with the bicycle? Is there a part (b)?

Car and bicycle collision problem

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