Comparing Distances in Sport & Automobiles

[Rylynn97]

1. A sport car starting from rest can attain a speed of 60 mi/hr in 8s. A runner can do a 100-yd dash in 9.8s. Assume that the runner is moving with a constant speed and that the car starts at the instant he passes it. How far will both travel until the car overtakes the runner? (convert distances to ft, time to s)

2. An automobile traveling 56 km/hr is 24m from a barrier when the driver slams on the brakes. The car hits the barrier two seconds later. (a) What was the car's deceleration before the impact? (b) How fast was the car traveling at impact? (distances to m, time to s)

Help?

 

[dimensionless]

1.

What is the cars acceleration? Find equations for both the positions of both the car and the runner as functions of time. They should intersect at t=0 and t= t2, where t2 is the time at which the car overtakes the runner. Then find how far the runner has traveled at t=t2.

2.

Given that the decelleration is a, integrate it to find an equation for the velocity v(t) that meets the initial condition of v(0) = 56 km/hr. Integrate the velocity equation to get an equation for the distance x(t) that meets the boundary conditions x(0) = 0 and x(2s) = 24 m.

 

[m2003]

I would approach this question by first converting all units to the standard SI units of meters, seconds, and meters per second. This will allow for a more accurate and consistent comparison between the two scenarios.

1. To convert the speed of the sport car from miles per hour to meters per second, we can use the conversion factor 1 mi/hr = 0.44704 m/s. Therefore, the car's speed is 26.8 m/s (60 mi/hr * 0.44704 m/s). Similarly, to convert the distance of the 100-yd dash to meters, we can use the conversion factor 1 yd = 0.9144 m. Therefore, the distance covered by the runner is 91.44 m (100 yd * 0.9144 m/yd).

Now, we can use the equation d = vt to calculate the distance traveled by both the car and runner. For the car, d = (26.8 m/s)(8 s) = 214.4 m. For the runner, d = (91.44 m)(9.8 s) = 896.512 m. This means that the car will overtake the runner after traveling a distance of 214.4 m, which is significantly shorter than the distance covered by the runner.

2. To convert the speed of the car from kilometers per hour to meters per second, we can use the conversion factor 1 km/hr = 0.277778 m/s. Therefore, the car's speed is 15.555 m/s (56 km/hr * 0.277778 m/s). We can also convert the distance of 24 m to meters.

(a) To calculate the car's deceleration, we can use the equation a = (vf - vi)/t, where vf is the final velocity, vi is the initial velocity, and t is the time. We know that the final velocity is 0 m/s (since the car hits the barrier) and the initial velocity is 15.555 m/s. The time can be calculated by using the fact that the car hits the barrier two seconds later, so t = 2 s. Therefore, the deceleration is -7.7775 m/s^2.

(b) To calculate the car's speed at impact, we can use the equation vf = vi + at. We know that vi is 15.555 m/s