Question

A motorcycle is following a car that is traveling at constant speed on a straight highway. Initially, the car and the motorcycle are both traveling at the same speed of 19.5 m/s , and the distance between them is 86.0 m . After t1 = 2.00 s , the motorcycle starts to accelerate at a rate of 7.00 m/s2 . The motorcycle catches up with the car at some time t2.

How long does it take from the moment when the motorcycle starts to accelerate until it catches up with the car? In other words, find t2-t1.
How far does the motorcycle travel from the moment it starts to accelerate (at time T1) until it catches up with the car (at time t2)? Should you need to use an answer from a previous part, make sure you use the unrounded value.

Answer 1

To find the time it takes for the motorcycle to catch up with the car, we can start by determining the position of the car and the motorcycle at time t1 and time t2.

Given:

Initial speed of the car and motorcycle (v0) = 19.5 m/s

Distance between the car and motorcycle at t1 (d) = 86.0 m

Acceleration of the motorcycle (a) = 7.00 m/s²

Time when the motorcycle starts to accelerate (t1) = 2.00 s

First, let's find the position of the car and motorcycle at time t1. The distance traveled by both can be calculated using the equation:

d = v0 * t

For the car:

d_car = v0 * t1

For the motorcycle:

d_motorcycle = v0 * t1

Next, let's find the time it takes for the motorcycle to catch up with the car (t2). We'll use the following kinematic equation:

d = v0 * t + (1/2) * a * t²

We know the distance between the car and motorcycle at t1, which is 86.0 m. We'll set up the equation for the motorcycle:

86.0 = v0 * t2 + (1/2) * a * t2²

Now, let's solve for t2. Rearranging the equation, we have:

(1/2) * a * t2² + v0 * t2 - 86.0 = 0

Using the quadratic formula, we can find t2:

t2 = (-v0 ± sqrt(v0² - 4 * (1/2) * a * (-86.0))) / (2 * (1/2) * a)

t2 = (-19.5 ± sqrt(19.5² - 2 * 7.00 * (-86.0))) / (2 * 7.00)

After solving this equation, we get two possible values for t2, but we need to select the positive value since time cannot be negative.

Now that we have t2, we can find the distance traveled by the motorcycle from t1 to t2. We'll use the equation:

d = v0 * t + (1/2) * a * t²

For the motorcycle:

d_motorcycle = v0 * (t2 - t1) + (1/2) * a * (t2 - t1)²

Plug in the values of v0, a, t1, and t2 to calculate the distance traveled by the motorcycle.

Please provide the values of v0, d, t1, a, and the results can be calculated.