Question related to motion / kinematics

[hasnainzeenwa]

Homework Statement

A bus starts moving with an acceleration of 2m/s^2. a cyclist, 96m behind the bus starts moving towards the bus simultaneously at 20m/s. in how much time will he overtake the bus.

Kindly point out the mistake in my solution too.

Homework Equations

v = u + at
s = ut + 1/2 * at^2

The Attempt at a Solution

As far as my thinking takes me, we have to use the concept of subtracting 2 velocities but as this is accelerated motion what I've done is that written a function f(t)

f(t) = u + at

in this case the initial velocity is 0

therefore, f(t) = at

which will give you the instantaneous velocity.

The cyclist is 96m behind therefore we need to find the time it would take for him to cover 96m
so I've come up with the following eqn.

(20 - f(t)) * t = 96

(20 - 2t) * t = 96

20t - 2t^2 = 96

We now have a quadratic equation after solving it I get the answer 5, but that is wrong and I don't know where I'm making a mistake, please point that out too.

 

[Doc Al]

(20 - f(t)) * t = 96

This assumes that the speed is constant, but it's not.

Try this:
Write the position of the cyclist as a function of time. Write the position of the bus as a function of time. Solve for when those positions are equal.

 

[hasnainzeenwa]

"This assumes that the speed is constant."

I didn't understand how does it assume that.

 

[Doc Al]

"This assumes that the speed is constant."

I didn't understand how does it assume that.

You used the equation D = V*t, which applies to constant speed motion. (Even when you replace V with a function of time.)

Example:

Say that V = at.

What's the distance as a function of time? Using V*t, you'll get D = at², which is incorrect. As you know, it is D = 1/2 at².

 

[Nickie]

Your approach to solving this problem is correct. However, there is a small mistake in your quadratic equation.

The correct equation should be:
20t - 2t^2 = 96

You have written:
(20 - 2t) * t = 96

This mistake changes the coefficient of t^2 from -2 to -1, which leads to an incorrect solution.

Correcting this mistake, the quadratic equation becomes:
-2t^2 + 20t - 96 = 0

Solving this equation will give you the correct answer of t = 6 seconds.

Overall, your understanding and approach to solving this problem using the equations of motion is correct. Keep practicing and double-checking your calculations to avoid small mistakes like this.