Sprinting Question Involving Constant Acceleration

[Emethyst]

Homework Statement

The world record for the 100m dash is 9.58s, and for the 60m dash it is 6.38s. Assume a sprinter accelerates at a constant rate up to a maximum velocity that is maintained for the rest of the race, no matter how long it is.

a) Find the acceleration (both sprinters will have the same acceleration)
b) Find the duration of this acceleration
c) Find the maximum velocity achieved by this acceleration
d) The world record for the 200m dash is 19.19s, while for the 1000m run it is 131.96s. Are these times consistent with the assumptions made?

Homework Equations

Kinematic equations

The Attempt at a Solution

This question is really stumping me. I know that I have to link the two values given to find the acceleration in part a, and thus everything else asked for, but I'm not sure how to go about doing this. My guess was to first find the each of the accelerations for the values given and then average them, but this does not seem to be the right way to solve for it. If anyone can point me in the right direction here it would be greatly appreciated, thanks in advance.

 

[method_man]

Homework Statement

The world record for the 100m dash is 9.58s, and for the 60m dash it is 6.38s. Assume a sprinter accelerates at a constant rate up to a maximum velocity that is maintained for the rest of the race, no matter how long it is.

a) Find the acceleration (both sprinters will have the same acceleration)
b) Find the duration of this acceleration
c) Find the maximum velocity achieved by this acceleration
d) The world record for the 200m dash is 19.19s, while for the 1000m run it is 131.96s. Are these times consistent with the assumptions made?

Homework Equations

Kinematic equations

The Attempt at a Solution

This question is really stumping me. I know that I have to link the two values given to find the acceleration in part a, and thus everything else asked for, but I'm not sure how to go about doing this. My guess was to first find the each of the accelerations for the values given and then average them, but this does not seem to be the right way to solve for it. If anyone can point me in the right direction here it would be greatly appreciated, thanks in advance.

Hy. I would like to point you on this key sentence

Assume a sprinter accelerates at a constant rate up to a maximum velocity that is maintained for the rest of the race

O.K. So, each displacement consists of 2 time intervals. First interval is when they accelerate, and the second one is when they run with constant speed.
Your displacement formula would be s=1/2*a*t²+v_max*t₁
This formula you must use for 100 m and for 60 m race. You know that v_max=a*t. Time t is equal for both races, and time t₁ will be different. Total time which is given for both races is equal to t+t₁.

 

[tomcue]

I would approach this question by first identifying the variables given and their relationship to each other. The two times provided (9.58s for 100m and 6.38s for 60m) represent the time it takes for the sprinters to cover the respective distances. We can use the kinematic equation d = v0t + 1/2at^2 (where d is distance, v0 is initial velocity, a is acceleration, and t is time) to find the acceleration for each sprinter.

For the 100m dash, we can set d = 100m and t = 9.58s. Assuming that the initial velocity is 0 (since the sprinter starts from a stationary position), we can solve for a:

100m = 0 + 1/2a(9.58s)^2
a = 4.21 m/s^2

Similarly, for the 60m dash, we can set d = 60m and t = 6.38s, and solve for a:

60m = 0 + 1/2a(6.38s)^2
a = 5.93 m/s^2

Since both sprinters are assumed to have the same acceleration, we can take the average of these two values to find the value of a:

(a for 100m + a for 60m) / 2 = (4.21 + 5.93) / 2 = 5.07 m/s^2

This is the acceleration that both sprinters would have in order to achieve their respective world records.

For part b, we can use the kinematic equation v = v0 + at (where v is final velocity, v0 is initial velocity, a is acceleration, and t is time) to find the duration of this acceleration. We know that the final velocity is the maximum velocity achieved, and the initial velocity is 0. So we can set v = maximum velocity and v0 = 0, and solve for t:

maximum velocity = 0 + 5.07 m/s^2 * t
t = maximum velocity / 5.07 m/s^2

For part c, we can simply substitute the value of t we found in part b into the same equation, this time solving for the maximum velocity:

maximum velocity = 5.07 m/s^2 * (maximum